Results of Elementary Functions

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
4.2.E8 log a ⁑ z = ln ⁑ z / ln ⁑ a π‘Ž 𝑧 𝑧 π‘Ž {\displaystyle{\displaystyle\operatorname{log}_{a}z=\ifrac{\ln z}{\ln a}}} log[a](z)=(ln(z))/(ln(a)) Log[a,z]=Divide[Log[z],Log[a]] Successful Successful - -
4.2.E9 log a ⁑ z = log b ⁑ z log b ⁑ a π‘Ž 𝑧 𝑏 𝑧 𝑏 π‘Ž {\displaystyle{\displaystyle\operatorname{log}_{a}z=\frac{\operatorname{log}_{% b}z}{\operatorname{log}_{b}a}}} log[a](z)=(log[b](z))/(log[b](a)) Log[a,z]=Divide[Log[b,z],Log[b,a]] Successful Successful - -
4.2.E10 log a ⁑ b = 1 log b ⁑ a π‘Ž 𝑏 1 𝑏 π‘Ž {\displaystyle{\displaystyle\operatorname{log}_{a}b=\frac{1}{\operatorname{log% }_{b}a}}} log[a](b)=(1)/(log[b](a)) Log[a,b]=Divide[1,Log[b,a]] Successful Successful - -
4.2.E12 ln ⁑ e = 1 𝑒 1 {\displaystyle{\displaystyle\ln e=1}} ln(exp(1))= 1 Log[E]= 1 Successful Successful - -
4.2.E13 ∫ 1 e d t t = 1 superscript subscript 1 𝑒 𝑑 𝑑 1 {\displaystyle{\displaystyle\int_{1}^{e}\frac{\mathrm{d}t}{t}=1}} int((1)/(t), t = 1..exp(1))= 1 Integrate[Divide[1,t], {t, 1, E}]= 1 Successful Successful - -
4.2.E14 log e ⁑ z = ln ⁑ z 𝑒 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{log}_{e}z=\ln z}} log[exp(1)](z)= ln(z) Log[E,z]= Log[z] Successful Successful - -
4.2.E15 log 10 ⁑ z = ( ln ⁑ z ) / ( ln ⁑ 10 ) 10 𝑧 𝑧 10 {\displaystyle{\displaystyle\operatorname{log}_{10}z=\ifrac{(\ln z)}{(\ln 10)}}} log[10](z)=(ln(z))/(ln(10)) Log[10,z]=Divide[Log[z],Log[10]] Successful Successful - -
4.2.E15 ( ln ⁑ z ) / ( ln ⁑ 10 ) = ( log 10 ⁑ e ) ⁒ ln ⁑ z 𝑧 10 10 𝑒 𝑧 {\displaystyle{\displaystyle\ifrac{(\ln z)}{(\ln 10)}=(\operatorname{log}_{10}% e)\ln z}} (ln(z))/(ln(10))=(log[10](exp(1)))* ln(z) Divide[Log[z],Log[10]]=(Log[10,E])* Log[z] Successful Successful - -
4.2.E16 ln ⁑ z = ( ln ⁑ 10 ) ⁒ log 10 ⁑ z 𝑧 10 10 𝑧 {\displaystyle{\displaystyle\ln z=(\ln 10)\operatorname{log}_{10}z}} ln(z)=(ln(10))* log[10](z) Log[z]=(Log[10])* Log[10,z] Successful Successful - -
4.2.E20 exp ⁑ ( z + 2 ⁒ Ο€ ⁒ i ) = exp ⁑ z 𝑧 2 πœ‹ 𝑖 𝑧 {\displaystyle{\displaystyle\exp\left(z+2\pi i\right)=\exp z}} exp(z + 2*Pi*I)= exp(z) Exp[z + 2*Pi*I]= Exp[z] Successful Successful - -
4.2.E21 exp ⁑ ( - z ) = 1 / exp ⁑ ( z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\exp\left(-z\right)=1/\exp\left(z\right)}} exp(- z)= 1/ exp(z) Exp[- z]= 1/ Exp[z] Successful Successful - -
4.2.E22 | exp ⁑ z | = exp ⁑ ( β„œ ⁑ z ) 𝑧 𝑧 {\displaystyle{\displaystyle|\exp z|=\exp\left(\Re z\right)}} abs(exp(z))= exp(Re(z)) Abs[Exp[z]]= Exp[Re[z]] Successful Successful - -
4.2.E23 ph ⁑ ( exp ⁑ z ) = β„‘ ⁑ z + 2 ⁒ k ⁒ Ο€ phase 𝑧 𝑧 2 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{ph}\left(\exp z\right)=\Im z+2k\pi}} argument(exp(z))= Im(z)+ 2*k*Pi Arg[Exp[z]]= Im[z]+ 2*k*Pi Failure Failure
Fail
-18.84955592 <- {z = 2^(1/2)+I*2^(1/2), k = 3}
-18.84955592 <- {z = 2^(1/2)-I*2^(1/2), k = 3}
-18.84955592 <- {z = -2^(1/2)-I*2^(1/2), k = 3}
-18.84955592 <- {z = -2^(1/2)+I*2^(1/2), k = 3}
Fail
-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.2.E24 exp ⁑ z = e x ⁒ cos ⁑ y + i ⁒ e x ⁒ sin ⁑ y 𝑧 superscript 𝑒 π‘₯ 𝑦 𝑖 superscript 𝑒 π‘₯ 𝑦 {\displaystyle{\displaystyle\exp z=e^{x}\cos y+ie^{x}\sin y}} exp(z)= exp(x)*cos(y)+ I*exp(x)*sin(y) Exp[z]= Exp[x]*Cos[y]+ I*Exp[x]*Sin[y] Failure Failure
Fail
-.8272584772+1.775573363*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
1.772639846+1.591201978*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
3.332514076+3.679324696*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-3.350888586-2.154747662*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[-0.8272584783533998, 1.7755733643246545] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.7726398453192989, 1.591201979498678] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.332514075382279, 3.679324697962366] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.3508885868787868, -2.154747660864471] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.2.E26 z a = exp ⁑ ( a ⁒ Ln ⁑ z ) superscript 𝑧 π‘Ž π‘Ž multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle z^{a}=\exp\left(a\operatorname{Ln}z\right)}} (z)^(a)= exp(a*ln(z)) (z)^(a)= Exp[a*Log[z]] Successful Failure - Successful
4.2.E28 z a = exp ⁑ ( a ⁒ ln ⁑ z ) superscript 𝑧 π‘Ž π‘Ž 𝑧 {\displaystyle{\displaystyle z^{a}=\exp\left(a\ln z\right)}} (z)^(a)= exp(a*ln(z)) (z)^(a)= Exp[a*Log[z]] Successful Successful - -
4.2.E29 | z a | = | z | β„œ ⁑ a ⁒ exp ⁑ ( - ( β„‘ ⁑ a ) ⁒ ph ⁑ z ) superscript 𝑧 π‘Ž superscript 𝑧 π‘Ž π‘Ž phase 𝑧 {\displaystyle{\displaystyle|z^{a}|=|z|^{\Re a}\exp\left(-(\Im a)\operatorname% {ph}z\right)}} abs((z)^(a))=(abs(z))^(Re(a))* exp(-(Im(a))* argument(z)) Abs[(z)^(a)]=(Abs[z])^(Re[a])* Exp[-(Im[a])* Arg[z]] Failure Failure Successful Successful
4.2.E30 ph ⁑ ( z a ) = ( β„œ ⁑ a ) ⁒ ph ⁑ z + ( β„‘ ⁑ a ) ⁒ ln ⁑ | z | phase superscript 𝑧 π‘Ž π‘Ž phase 𝑧 π‘Ž 𝑧 {\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=(\Re a)% \operatorname{ph}z+(\Im a)\ln|z|}} argument((z)^(a))=(Re(a))* argument(z)+(Im(a))* ln(abs(z)) Arg[(z)^(a)]=(Re[a])* Arg[z]+(Im[a])* Log[Abs[z]] Failure Failure
Fail
-6.283185309 <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
6.283185309 <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
6.283185309 <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
-6.283185309 <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
Fail
-6.283185307179586 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
6.283185307179586 <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
6.283185307179586 <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
-6.283185307179586 <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
4.2#Ex2 ph ⁑ ( z a ) = a ⁒ ph ⁑ z phase superscript 𝑧 π‘Ž π‘Ž phase 𝑧 {\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=a\operatorname% {ph}z}} argument((z)^(a))= a*argument(z) Arg[(z)^(a)]= a*Arg[z] Failure Failure
Fail
.980258143-1.110720734*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.9802581426+1.110720734*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
.980258144+3.332162204*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-5.302927166-3.332162204*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.9802581434685473, -1.1107207345395915] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.9802581434685472, 1.1107207345395915] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.9802581434685469, 3.332162203618774] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.302927163711039, -3.332162203618774] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.2.E32 e z = exp ⁑ z superscript 𝑒 𝑧 𝑧 {\displaystyle{\displaystyle e^{z}=\exp z}} exp(z)= exp(z) Exp[z]= Exp[z] Successful Successful - -
4.2.E33 e z = ( exp ⁑ z ) ⁒ exp ⁑ ( 2 ⁒ k ⁒ z ⁒ Ο€ ⁒ i ) superscript 𝑒 𝑧 𝑧 2 π‘˜ 𝑧 πœ‹ imaginary-unit {\displaystyle{\displaystyle e^{z}=(\exp z)\exp\left(2kz\pi\mathrm{i}\right)}} exp(z)=(exp(z))* exp(2*k*z*Pi*I) Exp[z]=(Exp[z])* Exp[2*k*z*Pi*I] Failure Failure
Fail
.6414354628+4.062928650*I <- {z = 2^(1/2)+I*2^(1/2), k = 3}
-.1544020768e13-.1710664597e12*I <- {z = 2^(1/2)-I*2^(1/2), k = 3}
.8993679173e11+.1849553851e11*I <- {z = -2^(1/2)-I*2^(1/2), k = 3}
.3791252193e-1+.2401424313*I <- {z = -2^(1/2)+I*2^(1/2), k = 3}
Fail
Complex[0.6414354615731531, 4.062928651501303] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.5440207807554412*^12, -1.710664745395911*^11] <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[8.993679264986926*^10, 1.8495537828408436*^10] <- {Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.03791252182632387, 0.24014243117514714] <- {Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.2.E36 - Ο€ ≀ β„‘ ⁑ ( 1 a ⁒ Ln ⁑ w ) πœ‹ 1 π‘Ž multivalued-natural-logarithm 𝑀 {\displaystyle{\displaystyle-\pi<=\Im\left(\frac{1}{a}\operatorname{Ln}w\right% )}} - Pi < = Im((1)/(a)*ln(w)) - Pi < = Im[Divide[1,a]*Log[w]] Failure Failure Successful Successful
4.2.E36 β„‘ ⁑ ( 1 a ⁒ Ln ⁑ w ) ≀ Ο€ 1 π‘Ž multivalued-natural-logarithm 𝑀 πœ‹ {\displaystyle{\displaystyle\Im\left(\frac{1}{a}\operatorname{Ln}w\right)<=\pi}} Im((1)/(a)*ln(w))< = Pi Im[Divide[1,a]*Log[w]]< = Pi Failure Failure Successful Successful
4.4.E1 ln ⁑ 1 = 0 1 0 {\displaystyle{\displaystyle\ln 1=0}} ln(1)= 0 Log[1]= 0 Successful Successful - -
4.4.E2 ln ⁑ ( - 1 + i ⁒ 0 ) = + Ο€ ⁒ i 1 imaginary-unit 0 πœ‹ imaginary-unit {\displaystyle{\displaystyle\ln\left(-1+\mathrm{i}0\right)=+\pi\mathrm{i}}} ln(- 1 + I*0)= + Pi*I Log[- 1 + I*0]= + Pi*I Successful Successful - -
4.4.E2 ln ⁑ ( - 1 - i ⁒ 0 ) = - Ο€ ⁒ i 1 imaginary-unit 0 πœ‹ imaginary-unit {\displaystyle{\displaystyle\ln\left(-1-\mathrm{i}0\right)=-\pi\mathrm{i}}} ln(- 1 - I*0)= - Pi*I Log[- 1 - I*0]= - Pi*I Failure Failure
Fail
6.283185308*I <- {}
Fail
Complex[0.0, 6.283185307179586] <- {}
4.4.E3 ln ⁑ ( + i ) = + 1 2 ⁒ Ο€ ⁒ i imaginary-unit 1 2 πœ‹ imaginary-unit {\displaystyle{\displaystyle\ln\left(+\mathrm{i}\right)=+\tfrac{1}{2}\pi% \mathrm{i}}} ln(+ I)= +(1)/(2)*Pi*I Log[+ I]= +Divide[1,2]*Pi*I Successful Successful - -
4.4.E3 ln ⁑ ( - i ) = - 1 2 ⁒ Ο€ ⁒ i imaginary-unit 1 2 πœ‹ imaginary-unit {\displaystyle{\displaystyle\ln\left(-\mathrm{i}\right)=-\tfrac{1}{2}\pi% \mathrm{i}}} ln(- I)= -(1)/(2)*Pi*I Log[- I]= -Divide[1,2]*Pi*I Successful Successful - -
4.4.E5 e + Ο€ ⁒ i = - 1 superscript 𝑒 πœ‹ imaginary-unit 1 {\displaystyle{\displaystyle e^{+\pi\mathrm{i}}=-1}} exp(+ Pi*I)= - 1 Exp[+ Pi*I]= - 1 Successful Successful - -
4.4.E5 e - Ο€ ⁒ i = - 1 superscript 𝑒 πœ‹ imaginary-unit 1 {\displaystyle{\displaystyle e^{-\pi\mathrm{i}}=-1}} exp(- Pi*I)= - 1 Exp[- Pi*I]= - 1 Successful Successful - -
4.4.E6 e + Ο€ ⁒ i / 2 = + i superscript 𝑒 πœ‹ imaginary-unit 2 imaginary-unit {\displaystyle{\displaystyle e^{+\pi\mathrm{i}/2}=+\mathrm{i}}} exp(+ Pi*I/ 2)= + I Exp[+ Pi*I/ 2]= + I Successful Successful - -
4.4.E6 e - Ο€ ⁒ i / 2 = - i superscript 𝑒 πœ‹ imaginary-unit 2 imaginary-unit {\displaystyle{\displaystyle e^{-\pi\mathrm{i}/2}=-\mathrm{i}}} exp(- Pi*I/ 2)= - I Exp[- Pi*I/ 2]= - I Successful Successful - -
4.4.E7 e 2 ⁒ Ο€ ⁒ k ⁒ i = 1 superscript 𝑒 2 πœ‹ π‘˜ imaginary-unit 1 {\displaystyle{\displaystyle e^{2\pi k\mathrm{i}}=1}} exp(2*Pi*k*I)= 1 Exp[2*Pi*k*I]= 1 Successful Failure - Successful
4.4.E8 e + Ο€ ⁒ i / 3 = 1 2 + i ⁒ 3 2 superscript 𝑒 πœ‹ imaginary-unit 3 1 2 imaginary-unit 3 2 {\displaystyle{\displaystyle e^{+\pi\mathrm{i}/3}=\frac{1}{2}+\mathrm{i}\frac{% \sqrt{3}}{2}}} exp(+ Pi*I/ 3)=(1)/(2)+ I*(sqrt(3))/(2) Exp[+ Pi*I/ 3]=Divide[1,2]+ I*Divide[Sqrt[3],2] Successful Successful - -
4.4.E8 e - Ο€ ⁒ i / 3 = 1 2 - i ⁒ 3 2 superscript 𝑒 πœ‹ imaginary-unit 3 1 2 imaginary-unit 3 2 {\displaystyle{\displaystyle e^{-\pi\mathrm{i}/3}=\frac{1}{2}-\mathrm{i}\frac{% \sqrt{3}}{2}}} exp(- Pi*I/ 3)=(1)/(2)- I*(sqrt(3))/(2) Exp[- Pi*I/ 3]=Divide[1,2]- I*Divide[Sqrt[3],2] Successful Successful - -
4.4.E9 e + 2 ⁒ Ο€ ⁒ i / 3 = - 1 2 + i ⁒ 3 2 superscript 𝑒 2 πœ‹ imaginary-unit 3 1 2 imaginary-unit 3 2 {\displaystyle{\displaystyle e^{+2\pi\mathrm{i}/3}=-\frac{1}{2}+\mathrm{i}% \frac{\sqrt{3}}{2}}} exp(+ 2*Pi*I/ 3)= -(1)/(2)+ I*(sqrt(3))/(2) Exp[+ 2*Pi*I/ 3]= -Divide[1,2]+ I*Divide[Sqrt[3],2] Successful Successful - -
4.4.E9 e - 2 ⁒ Ο€ ⁒ i / 3 = - 1 2 - i ⁒ 3 2 superscript 𝑒 2 πœ‹ imaginary-unit 3 1 2 imaginary-unit 3 2 {\displaystyle{\displaystyle e^{-2\pi\mathrm{i}/3}=-\frac{1}{2}-\mathrm{i}% \frac{\sqrt{3}}{2}}} exp(- 2*Pi*I/ 3)= -(1)/(2)- I*(sqrt(3))/(2) Exp[- 2*Pi*I/ 3]= -Divide[1,2]- I*Divide[Sqrt[3],2] Successful Successful - -
4.4.E10 e + Ο€ ⁒ i / 4 = 1 2 + i ⁒ 1 2 superscript 𝑒 πœ‹ imaginary-unit 4 1 2 imaginary-unit 1 2 {\displaystyle{\displaystyle e^{+\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}+\mathrm{i% }\frac{1}{\sqrt{2}}}} exp(+ Pi*I/ 4)=(1)/(sqrt(2))+ I*(1)/(sqrt(2)) Exp[+ Pi*I/ 4]=Divide[1,Sqrt[2]]+ I*Divide[1,Sqrt[2]] Successful Successful - -
4.4.E10 e - Ο€ ⁒ i / 4 = 1 2 - i ⁒ 1 2 superscript 𝑒 πœ‹ imaginary-unit 4 1 2 imaginary-unit 1 2 {\displaystyle{\displaystyle e^{-\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}-\mathrm{i% }\frac{1}{\sqrt{2}}}} exp(- Pi*I/ 4)=(1)/(sqrt(2))- I*(1)/(sqrt(2)) Exp[- Pi*I/ 4]=Divide[1,Sqrt[2]]- I*Divide[1,Sqrt[2]] Successful Successful - -
4.4.E11 e + 3 ⁒ Ο€ ⁒ i / 4 = - 1 2 + i ⁒ 1 2 superscript 𝑒 3 πœ‹ imaginary-unit 4 1 2 imaginary-unit 1 2 {\displaystyle{\displaystyle e^{+3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}+\mathrm% {i}\frac{1}{\sqrt{2}}}} exp(+ 3*Pi*I/ 4)= -(1)/(sqrt(2))+ I*(1)/(sqrt(2)) Exp[+ 3*Pi*I/ 4]= -Divide[1,Sqrt[2]]+ I*Divide[1,Sqrt[2]] Successful Successful - -
4.4.E11 e - 3 ⁒ Ο€ ⁒ i / 4 = - 1 2 - i ⁒ 1 2 superscript 𝑒 3 πœ‹ imaginary-unit 4 1 2 imaginary-unit 1 2 {\displaystyle{\displaystyle e^{-3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}-\mathrm% {i}\frac{1}{\sqrt{2}}}} exp(- 3*Pi*I/ 4)= -(1)/(sqrt(2))- I*(1)/(sqrt(2)) Exp[- 3*Pi*I/ 4]= -Divide[1,Sqrt[2]]- I*Divide[1,Sqrt[2]] Successful Successful - -
4.4.E12 i + i = e - Ο€ / 2 imaginary-unit imaginary-unit superscript 𝑒 πœ‹ 2 {\displaystyle{\displaystyle{\mathrm{i}^{+\mathrm{i}}}=e^{-\pi/2}}} (I)^(+ I)= exp(- Pi/ 2) (I)^(+ I)= Exp[- Pi/ 2] Successful Successful - -
4.4.E12 i - i = e + Ο€ / 2 imaginary-unit imaginary-unit superscript 𝑒 πœ‹ 2 {\displaystyle{\displaystyle{\mathrm{i}^{-\mathrm{i}}}=e^{+\pi/2}}} (I)^(- I)= exp(+ Pi/ 2) (I)^(- I)= Exp[+ Pi/ 2] Successful Successful - -
4.4.E13 lim x β†’ ∞ ⁑ x - a ⁒ ln ⁑ x = 0 subscript β†’ π‘₯ superscript π‘₯ π‘Ž π‘₯ 0 {\displaystyle{\displaystyle\lim_{x\to\infty}x^{-a}\ln x=0}} limit((x)^(- a)* ln(x), x = infinity)= 0 Limit[(x)^(- a)* Log[x], x -> Infinity]= 0 Successful Failure - Successful
4.4.E14 lim x β†’ 0 ⁑ x a ⁒ ln ⁑ x = 0 subscript β†’ π‘₯ 0 superscript π‘₯ π‘Ž π‘₯ 0 {\displaystyle{\displaystyle\lim_{x\to 0}x^{a}\ln x=0}} limit((x)^(a)* ln(x), x = 0)= 0 Limit[(x)^(a)* Log[x], x -> 0]= 0 Failure Failure Skip Successful
4.4.E19 lim n β†’ ∞ ⁑ ( ( βˆ‘ k = 1 n 1 k ) - ln ⁑ n ) = Ξ³ subscript β†’ 𝑛 subscript superscript 𝑛 π‘˜ 1 1 π‘˜ 𝑛 {\displaystyle{\displaystyle\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1% }{k}\right)-\ln n\right)=\gamma}} limit((sum((1)/(k), k = 1..n))- ln(n), n = infinity)= gamma Limit[(Sum[Divide[1,k], {k, 1, n}])- Log[n], n -> Infinity]= EulerGamma Successful Successful - -
4.5.E1 x 1 + x < ln ⁑ ( 1 + x ) π‘₯ 1 π‘₯ 1 π‘₯ {\displaystyle{\displaystyle\frac{x}{1+x}<\ln\left(1+x\right)}} (x)/(1 + x)< ln(1 + x) Divide[x,1 + x]< Log[1 + x] Failure Failure Skip Successful
4.5.E1 ln ⁑ ( 1 + x ) < x 1 π‘₯ π‘₯ {\displaystyle{\displaystyle\ln\left(1+x\right)<x}} ln(1 + x)< x Log[1 + x]< x Failure Failure Skip Successful
4.5.E2 x < - ln ⁑ ( 1 - x ) π‘₯ 1 π‘₯ {\displaystyle{\displaystyle x<-\ln\left(1-x\right)}} x < - ln(1 - x) x < - Log[1 - x] Failure Failure Skip Successful
4.5.E2 - ln ⁑ ( 1 - x ) < x 1 - x 1 π‘₯ π‘₯ 1 π‘₯ {\displaystyle{\displaystyle-\ln\left(1-x\right)<\frac{x}{1-x}}} - ln(1 - x)<(x)/(1 - x) - Log[1 - x]<Divide[x,1 - x] Failure Failure Skip Successful
4.5.E3 | ln ⁑ ( 1 - x ) | < 3 2 ⁒ x 1 π‘₯ 3 2 π‘₯ {\displaystyle{\displaystyle|\ln\left(1-x\right)|<\tfrac{3}{2}x}} abs(ln(1 - x))<(3)/(2)*x Abs[Log[1 - x]]<Divide[3,2]*x Failure Failure Error Successful
4.5.E4 ln ⁑ x ≀ x - 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle\ln x<=x-1}} ln(x)< = x - 1 Log[x]< = x - 1 Failure Failure Successful Successful
4.5.E5 ln ⁑ x ≀ a ⁒ ( x 1 / a - 1 ) π‘₯ π‘Ž superscript π‘₯ 1 π‘Ž 1 {\displaystyle{\displaystyle\ln x<=a(x^{1/a}-1)}} ln(x)< = a*((x)^(1/ a)- 1) Log[x]< = a*((x)^(1/ a)- 1) Failure Failure Successful Successful
4.5.E6 | ln ⁑ ( 1 + z ) | ≀ - ln ⁑ ( 1 - | z | ) 1 𝑧 1 𝑧 {\displaystyle{\displaystyle|\ln\left(1+z\right)|<=-\ln\left(1-|z|\right)}} abs(ln(1 + z))< = - ln(1 -abs(z)) Abs[Log[1 + z]]< = - Log[1 -Abs[z]] Failure Failure Successful Successful
4.7.E1 d d z ⁑ ln ⁑ z = 1 z derivative 𝑧 𝑧 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\ln z=\frac{1}{z}}} diff(ln(z), z)=(1)/(z) D[Log[z], z]=Divide[1,z] Successful Successful - -
4.7.E2 d d z ⁑ Ln ⁑ z = 1 z derivative 𝑧 multivalued-natural-logarithm 𝑧 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{Ln}z=% \frac{1}{z}}} diff(ln(z), z)=(1)/(z) D[Log[z], z]=Divide[1,z] Successful Successful - -
4.7.E3 d n d z n ⁑ ln ⁑ z = ( - 1 ) n - 1 ⁒ ( n - 1 ) ! ⁒ z - n derivative 𝑧 𝑛 𝑧 superscript 1 𝑛 1 𝑛 1 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\ln z=(-% 1)^{n-1}(n-1)!z^{-n}}} diff(ln(z), [z$(n)])=(- 1)^(n - 1)*factorial(n - 1)*(z)^(- n) D[Log[z], {z, n}]=(- 1)^(n - 1)*(n - 1)!*(z)^(- n) Failure Failure Successful Successful
4.7.E4 d n d z n ⁑ Ln ⁑ z = ( - 1 ) n - 1 ⁒ ( n - 1 ) ! ⁒ z - n derivative 𝑧 𝑛 multivalued-natural-logarithm 𝑧 superscript 1 𝑛 1 𝑛 1 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}% \operatorname{Ln}z=(-1)^{n-1}(n-1)!z^{-n}}} diff(ln(z), [z$(n)])=(- 1)^(n - 1)*factorial(n - 1)*(z)^(- n) D[Log[z], {z, n}]=(- 1)^(n - 1)*(n - 1)!*(z)^(- n) Failure Failure Successful Successful
4.7.E7 d d z ⁑ e z = e z derivative 𝑧 superscript 𝑒 𝑧 superscript 𝑒 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z}}} diff(exp(z), z)= exp(z) D[Exp[z], z]= Exp[z] Successful Successful - -
4.7.E8 d d z ⁑ e a ⁒ z = a ⁒ e a ⁒ z derivative 𝑧 superscript 𝑒 π‘Ž 𝑧 π‘Ž superscript 𝑒 π‘Ž 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}e^{az}=ae^{az}}} diff(exp(a*z), z)= a*exp(a*z) D[Exp[a*z], z]= a*Exp[a*z] Successful Successful - -
4.7.E9 d d z ⁑ a z = a z ⁒ ln ⁑ a derivative 𝑧 superscript π‘Ž 𝑧 superscript π‘Ž 𝑧 π‘Ž {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}a^{z}=a^{z}\ln a}} diff((a)^(z), z)= (a)^(z)* ln(a) D[(a)^(z), z]= (a)^(z)* Log[a] Successful Failure - Successful
4.7.E10 d d z ⁑ z a = a ⁒ z a - 1 derivative 𝑧 superscript 𝑧 π‘Ž π‘Ž superscript 𝑧 π‘Ž 1 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}z^{a}=az^{a-1}}} diff((z)^(a), z)= a*(z)^(a - 1) D[(z)^(a), z]= a*(z)^(a - 1) Successful Successful - -
4.7.E14 d 2 w d z 2 = a ⁒ w derivative 𝑀 𝑧 2 π‘Ž 𝑀 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=aw}} diff(w, [z$(2)])= a*w D[w, {z, 2}]= a*w Failure Failure Skip Successful
4.8.E1 Ln ⁑ ( z 1 ⁒ z 2 ) = Ln ⁑ z 1 + Ln ⁑ z 2 multivalued-natural-logarithm subscript 𝑧 1 subscript 𝑧 2 multivalued-natural-logarithm subscript 𝑧 1 multivalued-natural-logarithm subscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{Ln}\left(z_{1}z_{2}\right)=% \operatorname{Ln}z_{1}+\operatorname{Ln}z_{2}}} ln(z[1]*z[2])= ln(z[1])+ ln(z[2]) Log[Subscript[z, 1]*Subscript[z, 2]]= Log[Subscript[z, 1]]+ Log[Subscript[z, 2]] Failure Failure
Fail
.4e-9+6.283185307*I <- {z[1] = 2^(1/2)-I*2^(1/2), z[2] = -2^(1/2)-I*2^(1/2)}
.4e-9+6.283185307*I <- {z[1] = -2^(1/2)-I*2^(1/2), z[2] = 2^(1/2)-I*2^(1/2)}
0.+6.283185307*I <- {z[1] = -2^(1/2)-I*2^(1/2), z[2] = -2^(1/2)-I*2^(1/2)}
0.-6.283185307*I <- {z[1] = -2^(1/2)+I*2^(1/2), z[2] = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.8.E2 ln ⁑ ( z 1 ⁒ z 2 ) = ln ⁑ z 1 + ln ⁑ z 2 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\ln\left(z_{1}z_{2}\right)=\ln z_{1}+\ln z_{2}}} ln(z[1]*z[2])= ln(z[1])+ ln(z[2]) Log[Subscript[z, 1]*Subscript[z, 2]]= Log[Subscript[z, 1]]+ Log[Subscript[z, 2]] Failure Failure Skip
Fail
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
4.8.E3 Ln ⁑ z 1 z 2 = Ln ⁑ z 1 - Ln ⁑ z 2 multivalued-natural-logarithm subscript 𝑧 1 subscript 𝑧 2 multivalued-natural-logarithm subscript 𝑧 1 multivalued-natural-logarithm subscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname% {Ln}z_{1}-\operatorname{Ln}z_{2}}} ln((z[1])/(z[2]))= ln(z[1])- ln(z[2]) Log[Divide[Subscript[z, 1],Subscript[z, 2]]]= Log[Subscript[z, 1]]- Log[Subscript[z, 2]] Failure Failure
Fail
0.+6.283185307*I <- {z[1] = 2^(1/2)-I*2^(1/2), z[2] = -2^(1/2)+I*2^(1/2)}
0.+6.283185307*I <- {z[1] = -2^(1/2)-I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2)}
0.+6.283185307*I <- {z[1] = -2^(1/2)-I*2^(1/2), z[2] = -2^(1/2)+I*2^(1/2)}
0.-6.283185307*I <- {z[1] = -2^(1/2)+I*2^(1/2), z[2] = -2^(1/2)-I*2^(1/2)}
Fail
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
4.8.E4 ln ⁑ z 1 z 2 = ln ⁑ z 1 - ln ⁑ z 2 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\ln\frac{z_{1}}{z_{2}}=\ln z_{1}-\ln z_{2}}} ln((z[1])/(z[2]))= ln(z[1])- ln(z[2]) Log[Divide[Subscript[z, 1],Subscript[z, 2]]]= Log[Subscript[z, 1]]- Log[Subscript[z, 2]] Failure Failure Skip
Fail
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
4.8.E5 Ln ⁑ ( z n ) = n ⁒ Ln ⁑ z multivalued-natural-logarithm superscript 𝑧 𝑛 𝑛 multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\operatorname{Ln}\left(z^{n}\right)=n\operatorname% {Ln}z}} ln((z)^(n))= n*ln(z) Log[(z)^(n)]= n*Log[z] Failure Failure
Fail
0.+6.283185307*I <- {z = -2^(1/2)-I*2^(1/2), n = 3}
0.-6.283185307*I <- {z = -2^(1/2)+I*2^(1/2), n = 3}
Fail
Complex[0.0, 6.283185307179586] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -6.283185307179586] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.8.E6 ln ⁑ ( z n ) = n ⁒ ln ⁑ z superscript 𝑧 𝑛 𝑛 𝑧 {\displaystyle{\displaystyle\ln\left(z^{n}\right)=n\ln z}} ln((z)^(n))= n*ln(z) Log[(z)^(n)]= n*Log[z] Failure Failure Skip Successful
4.8.E7 ln ⁑ 1 z = - ln ⁑ z 1 𝑧 𝑧 {\displaystyle{\displaystyle\ln\frac{1}{z}=-\ln z}} ln((1)/(z))= - ln(z) Log[Divide[1,z]]= - Log[z] Failure Failure Skip Successful
4.8.E8 Ln ⁑ ( exp ⁑ z ) = z + 2 ⁒ k ⁒ Ο€ ⁒ i multivalued-natural-logarithm 𝑧 𝑧 2 π‘˜ πœ‹ imaginary-unit {\displaystyle{\displaystyle\operatorname{Ln}\left(\exp z\right)=z+2k\pi% \mathrm{i}}} ln(exp(z))= z + 2*k*Pi*I Log[Exp[z]]= z + 2*k*Pi*I Failure Failure
Fail
0.-18.84955592*I <- {z = 2^(1/2)+I*2^(1/2), k = 3}
0.-18.84955592*I <- {z = 2^(1/2)-I*2^(1/2), k = 3}
0.-18.84955592*I <- {z = -2^(1/2)-I*2^(1/2), k = 3}
0.-18.84955592*I <- {z = -2^(1/2)+I*2^(1/2), k = 3}
Fail
Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.8.E9 ln ⁑ ( exp ⁑ z ) = z 𝑧 𝑧 {\displaystyle{\displaystyle\ln\left(\exp z\right)=z}} ln(exp(z))= z Log[Exp[z]]= z Failure Failure Skip Successful
4.8.E10 exp ⁑ ( ln ⁑ z ) = exp ⁑ ( Ln ⁑ z ) 𝑧 multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\exp\left(\ln z\right)=\exp\left(\operatorname{Ln}% z\right)}} exp(ln(z))= exp(ln(z)) Exp[Log[z]]= Exp[Log[z]] Successful Successful - -
4.8.E10 exp ⁑ ( Ln ⁑ z ) = z multivalued-natural-logarithm 𝑧 𝑧 {\displaystyle{\displaystyle\exp\left(\operatorname{Ln}z\right)=z}} exp(ln(z))= z Exp[Log[z]]= z Successful Successful - -
4.8.E11 Ln ⁑ ( a z ) = z ⁒ Ln ⁑ a + 2 ⁒ k ⁒ Ο€ ⁒ i multivalued-natural-logarithm superscript π‘Ž 𝑧 𝑧 multivalued-natural-logarithm π‘Ž 2 π‘˜ πœ‹ imaginary-unit {\displaystyle{\displaystyle\operatorname{Ln}\left(a^{z}\right)=z\operatorname% {Ln}a+2k\pi\mathrm{i}}} ln((a)^(z))= z*ln(a)+ 2*k*Pi*I Log[(a)^(z)]= z*Log[a]+ 2*k*Pi*I Failure Failure
Fail
0.-18.84955592*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 3}
0.-18.84955592*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), k = 3}
0.-18.84955592*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), k = 3}
0.-18.84955592*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), k = 3}
... skip entries to safe data
Fail
Complex[1.1102230246251565*^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.3877787807814457*^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[4.440892098500626*^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.8.E12 ln ⁑ ( a z ) = z ⁒ ln ⁑ a + 2 ⁒ k ⁒ Ο€ ⁒ i superscript π‘Ž 𝑧 𝑧 π‘Ž 2 π‘˜ πœ‹ imaginary-unit {\displaystyle{\displaystyle\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i}}} ln((a)^(z))= z*ln(a)+ 2*k*Pi*I Log[(a)^(z)]= z*Log[a]+ 2*k*Pi*I Failure Failure
Fail
0.-6.283185308*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 1}
0.-12.56637062*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 2}
0.-18.84955592*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), k = 3}
0.-6.283185308*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
Fail
Complex[1.1102230246251565*^-16, -6.283185307179586] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.1102230246251565*^-16, -12.566370614359172] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.1102230246251565*^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -6.283185307179586] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.8.E13 ln ⁑ ( a x ) = x ⁒ ln ⁑ a superscript π‘Ž π‘₯ π‘₯ π‘Ž {\displaystyle{\displaystyle\ln\left(a^{x}\right)=x\ln a}} ln((a)^(x))= x*ln(a) Log[(a)^(x)]= x*Log[a] Failure Failure Successful Successful
4.10.E1 ∫ d z z = ln ⁑ z 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{z}=\ln z}} int((1)/(z), z)= ln(z) Integrate[Divide[1,z], z]= Log[z] Successful Successful - -
4.10.E2 ∫ ln ⁑ z ⁒ d z = z ⁒ ln ⁑ z - z 𝑧 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\int\ln z\mathrm{d}z=z\ln z-z}} int(ln(z), z)= z*ln(z)- z Integrate[Log[z], z]= z*Log[z]- z Successful Successful - -
4.10.E3 ∫ z n ⁒ ln ⁑ z ⁒ d z = z n + 1 n + 1 ⁒ ln ⁑ z - z n + 1 ( n + 1 ) 2 superscript 𝑧 𝑛 𝑧 𝑧 superscript 𝑧 𝑛 1 𝑛 1 𝑧 superscript 𝑧 𝑛 1 superscript 𝑛 1 2 {\displaystyle{\displaystyle\int z^{n}\ln z\mathrm{d}z=\frac{z^{n+1}}{n+1}\ln z% -\frac{z^{n+1}}{(n+1)^{2}}}} int((z)^(n)* ln(z), z)=((z)^(n + 1))/(n + 1)*ln(z)-((z)^(n + 1))/((n + 1)^(2)) Integrate[(z)^(n)* Log[z], z]=Divide[(z)^(n + 1),n + 1]*Log[z]-Divide[(z)^(n + 1),(n + 1)^(2)] Successful Successful - -
4.10.E4 ∫ d z z ⁒ ln ⁑ z = ln ⁑ ( ln ⁑ z ) 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{z\ln z}=\ln\left(\ln z% \right)}} int((1)/(z*ln(z)), z)= ln(ln(z)) Integrate[Divide[1,z*Log[z]], z]= Log[Log[z]] Successful Successful - -
4.10.E5 ∫ 0 1 ln ⁑ t 1 - t ⁒ d t = - Ο€ 2 6 superscript subscript 0 1 𝑑 1 𝑑 𝑑 superscript πœ‹ 2 6 {\displaystyle{\displaystyle\int_{0}^{1}\frac{\ln t}{1-t}\mathrm{d}t=-\frac{% \pi^{2}}{6}}} int((ln(t))/(1 - t), t = 0..1)= -((Pi)^(2))/(6) Integrate[Divide[Log[t],1 - t], {t, 0, 1}]= -Divide[(Pi)^(2),6] Successful Successful - -
4.10.E6 ∫ 0 1 ln ⁑ t 1 + t ⁒ d t = - Ο€ 2 12 superscript subscript 0 1 𝑑 1 𝑑 𝑑 superscript πœ‹ 2 12 {\displaystyle{\displaystyle\int_{0}^{1}\frac{\ln t}{1+t}\mathrm{d}t=-\frac{% \pi^{2}}{12}}} int((ln(t))/(1 + t), t = 0..1)= -((Pi)^(2))/(12) Integrate[Divide[Log[t],1 + t], {t, 0, 1}]= -Divide[(Pi)^(2),12] Successful Successful - -
4.10.E8 ∫ e a ⁒ z ⁒ d z = e a ⁒ z a superscript 𝑒 π‘Ž 𝑧 𝑧 superscript 𝑒 π‘Ž 𝑧 π‘Ž {\displaystyle{\displaystyle\int e^{az}\mathrm{d}z=\frac{e^{az}}{a}}} int(exp(a*z), z)=(exp(a*z))/(a) Integrate[Exp[a*z], z]=Divide[Exp[a*z],a] Successful Successful - -
4.10.E9 ∫ d z e a ⁒ z + b = 1 a ⁒ b ⁒ ( a ⁒ z - ln ⁑ ( e a ⁒ z + b ) ) 𝑧 superscript 𝑒 π‘Ž 𝑧 𝑏 1 π‘Ž 𝑏 π‘Ž 𝑧 superscript 𝑒 π‘Ž 𝑧 𝑏 {\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{e^{az}+b}=\frac{1}{ab}(az-% \ln\left(e^{az}+b\right))}} int((1)/(exp(a*z)+ b), z)=(1)/(a*b)*(a*z - ln(exp(a*z)+ b)) Integrate[Divide[1,Exp[a*z]+ b], z]=Divide[1,a*b]*(a*z - Log[Exp[a*z]+ b]) Failure Successful Skip -
4.10.E10 ∫ e a ⁒ z - 1 e a ⁒ z + 1 ⁒ d z = 2 a ⁒ ln ⁑ ( e a ⁒ z / 2 + e - a ⁒ z / 2 ) superscript 𝑒 π‘Ž 𝑧 1 superscript 𝑒 π‘Ž 𝑧 1 𝑧 2 π‘Ž superscript 𝑒 π‘Ž 𝑧 2 superscript 𝑒 π‘Ž 𝑧 2 {\displaystyle{\displaystyle\int\frac{e^{az}-1}{e^{az}+1}\mathrm{d}z=\frac{2}{% a}\ln\left(e^{az/2}+e^{-az/2}\right)}} int((exp(a*z)- 1)/(exp(a*z)+ 1), z)=(2)/(a)*ln(exp(a*z/ 2)+ exp(- a*z/ 2)) Integrate[Divide[Exp[a*z]- 1,Exp[a*z]+ 1], z]=Divide[2,a]*Log[Exp[a*z/ 2]+ Exp[- a*z/ 2]] Failure Failure Skip
Fail
Complex[-4.442882938158366, -4.442882938158366] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.442882938158366, -4.442882938158366] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.442882938158366, 4.442882938158366] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.442882938158366, 4.442882938158366] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
4.10.E11 ∫ - ∞ ∞ e - c ⁒ x 2 ⁒ d x = Ο€ c superscript subscript superscript 𝑒 𝑐 superscript π‘₯ 2 π‘₯ πœ‹ 𝑐 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-cx^{2}}\mathrm{d}x=% \sqrt{\frac{\pi}{c}}}} int(exp(- c*(x)^(2)), x = - infinity..infinity)=sqrt((Pi)/(c)) Integrate[Exp[- c*(x)^(2)], {x, - Infinity, Infinity}]=Sqrt[Divide[Pi,c]] Successful Failure - Skip
4.10.E12 ∫ 0 ln ⁑ 2 x ⁒ e x e x - 1 ⁒ d x = Ο€ 2 12 superscript subscript 0 2 π‘₯ superscript 𝑒 π‘₯ superscript 𝑒 π‘₯ 1 π‘₯ superscript πœ‹ 2 12 {\displaystyle{\displaystyle\int_{0}^{\ln 2}\frac{xe^{x}}{e^{x}-1}\mathrm{d}x=% \frac{\pi^{2}}{12}}} int((x*exp(x))/(exp(x)- 1), x = 0..ln(2))=((Pi)^(2))/(12) Integrate[Divide[x*Exp[x],Exp[x]- 1], {x, 0, Log[2]}]=Divide[(Pi)^(2),12] Successful Successful - -
4.10.E13 ∫ 0 ∞ d x e x + 1 = ln ⁑ 2 superscript subscript 0 π‘₯ superscript 𝑒 π‘₯ 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}x}{e^{x}+1}=\ln 2}} int((1)/(exp(x)+ 1), x = 0..infinity)= ln(2) Integrate[Divide[1,Exp[x]+ 1], {x, 0, Infinity}]= Log[2] Successful Successful - -
4.12.E6 Ο• ⁒ ( x ) = ln ⁑ ( x + 1 ) italic-Ο• π‘₯ π‘₯ 1 {\displaystyle{\displaystyle\phi(x)=\ln\left(x+1\right)}} phi*(x)= ln(x + 1) \[Phi]*(x)= Log[x + 1] Failure Failure Skip Successful
4.12.E9 ψ ⁒ ( x ) = β„“ + ln ( β„“ ) ⁑ x πœ“ π‘₯ β„“ β„“ π‘₯ {\displaystyle{\displaystyle\psi(x)=\ell+{\ln^{(\ell)}}x}} psi*(x)= ell + subs( temp=x, diff( ln(temp), temp$(ell) ) ) \[Psi]*(x)= \[ScriptL]+ (D[Log[temp], {temp, \[ScriptL]}]/.temp-> x) Failure Failure
Fail
.454653676+2.121320343*I <- {psi = 2^(1/2)+I*2^(1/2), ell = 1, x = 3/2}
.565764787+2.121320343*I <- {psi = 2^(1/2)+I*2^(1/2), ell = 2, x = 3/2}
-1.471272250+2.121320343*I <- {psi = 2^(1/2)+I*2^(1/2), ell = 3, x = 3/2}
.454653676-2.121320343*I <- {psi = 2^(1/2)-I*2^(1/2), ell = 1, x = 3/2}
... skip entries to safe data
Fail
Complex[0.45465367689297564, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[β„“, 1], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.5657647880040868, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[β„“, 2], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4712722490329502, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[β„“, 3], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.45465367689297564, -2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[β„“, 1], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.13.E1 W ⁒ e W = x π‘Š superscript 𝑒 π‘Š π‘₯ {\displaystyle{\displaystyle We^{W}=x}} W*exp(W)= x W*Exp[W]= x Failure Failure
Fail
-5.838722068+6.652975529*I <- {W = 2^(1/2)+I*2^(1/2), x = 1}
-6.838722068+6.652975529*I <- {W = 2^(1/2)+I*2^(1/2), x = 2}
-7.838722068+6.652975529*I <- {W = 2^(1/2)+I*2^(1/2), x = 3}
-5.838722068-6.652975529*I <- {W = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[-5.838722072781763, 6.652975531039188] <- {Rule[W, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[-6.838722072781763, 6.652975531039188] <- {Rule[W, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[-7.838722072781763, 6.652975531039188] <- {Rule[W, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[-5.838722072781763, -6.652975531039188] <- {Rule[W, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
... skip entries to safe data
4.13#Ex1 Wp ⁑ ( - 1 / e ) = Wm ⁑ ( - 1 / e ) Lambert-Wp 1 𝑒 Lambert-Wm 1 𝑒 {\displaystyle{\displaystyle\mathrm{Wp}\left(-1/e\right)=\mathrm{Wm}\left(-1/e% \right)}} LambertW(0, - 1/ exp(1))= LambertW(-1, - 1/ exp(1)) ProductLog[0, - 1/ E]= ProductLog[-1, - 1/ E] Successful Successful - -
4.13#Ex1 Wm ⁑ ( - 1 / e ) = - 1 Lambert-Wm 1 𝑒 1 {\displaystyle{\displaystyle\mathrm{Wm}\left(-1/e\right)=-1}} LambertW(-1, - 1/ exp(1))= - 1 ProductLog[-1, - 1/ E]= - 1 Successful Successful - -
4.13#Ex2 Wp ⁑ ( 0 ) = 0 Lambert-Wp 0 0 {\displaystyle{\displaystyle\mathrm{Wp}\left(0\right)=0}} LambertW(0, 0)= 0 ProductLog[0, 0]= 0 Successful Successful - -
4.13#Ex3 Wp ⁑ ( e ) = 1 Lambert-Wp 𝑒 1 {\displaystyle{\displaystyle\mathrm{Wp}\left(e\right)=1}} LambertW(0, exp(1))= 1 ProductLog[0, E]= 1 Successful Successful - -
4.13#Ex4 U + ln ⁑ U = x π‘ˆ π‘ˆ π‘₯ {\displaystyle{\displaystyle U+\ln U=x}} U + ln(U)= x U + Log[U]= x Failure Failure
Fail
1.107360742+2.199611725*I <- {U = 2^(1/2)+I*2^(1/2), x = 1}
.107360742+2.199611725*I <- {U = 2^(1/2)+I*2^(1/2), x = 2}
-.892639258+2.199611725*I <- {U = 2^(1/2)+I*2^(1/2), x = 3}
1.107360742-2.199611725*I <- {U = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[1.1073607429330403, 2.199611725770543] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[0.10736074293304043, 2.199611725770543] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[-0.8926392570669596, 2.199611725770543] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[1.1073607429330403, -2.199611725770543] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
... skip entries to safe data
4.13#Ex5 U = U ⁒ ( x ) π‘ˆ π‘ˆ π‘₯ {\displaystyle{\displaystyle U=U(x)}} U = U*(x) U = U*(x) Failure Failure
Fail
-1.414213562-1.414213562*I <- {U = 2^(1/2)+I*2^(1/2), x = 2}
-2.828427124-2.828427124*I <- {U = 2^(1/2)+I*2^(1/2), x = 3}
-1.414213562+1.414213562*I <- {U = 2^(1/2)-I*2^(1/2), x = 2}
-2.828427124+2.828427124*I <- {U = 2^(1/2)-I*2^(1/2), x = 3}
... skip entries to safe data
Fail
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[-2.8284271247461903, 2.8284271247461903] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
... skip entries to safe data
4.13#Ex5 U ⁒ ( x ) = W ⁑ ( e x ) π‘ˆ π‘₯ Lambert-W superscript 𝑒 π‘₯ {\displaystyle{\displaystyle U(x)=W\left(e^{x}\right)}} U*(x)= LambertW(exp(x)) U*(x)= ProductLog[Exp[x]] Failure Failure
Fail
.414213562+1.414213562*I <- {U = 2^(1/2)+I*2^(1/2), x = 1}
1.271281525+2.828427124*I <- {U = 2^(1/2)+I*2^(1/2), x = 2}
2.034700655+4.242640686*I <- {U = 2^(1/2)+I*2^(1/2), x = 3}
.414213562-1.414213562*I <- {U = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[1.2712815257485788, 2.8284271247461903] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[2.0347006555499627, 4.242640687119286] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
... skip entries to safe data
4.13.E4 d W d x = e - W 1 + W derivative Lambert-W π‘₯ superscript 𝑒 Lambert-W 1 Lambert-W {\displaystyle{\displaystyle\frac{\mathrm{d}W}{\mathrm{d}x}=\frac{e^{-W}}{1+W}}} diff(LambertW(x), =)*(exp(- LambertW($0)))/(1 + LambertW($0)) D[ProductLog[x], =]*Divide[Exp[- ProductLog[$0]],1 + ProductLog[$0]] Error Error - -
4.13.E5 Wp ⁑ ( x ) = βˆ‘ n = 1 ∞ ( - 1 ) n - 1 ⁒ n n - 2 ( n - 1 ) ! ⁒ x n Lambert-Wp π‘₯ superscript subscript 𝑛 1 superscript 1 𝑛 1 superscript 𝑛 𝑛 2 𝑛 1 superscript π‘₯ 𝑛 {\displaystyle{\displaystyle\mathrm{Wp}\left(x\right)=\sum_{n=1}^{\infty}(-1)^% {n-1}\frac{n^{n-2}}{(n-1)!}x^{n}}} LambertW(0, x)= sum((- 1)^(n - 1)*((n)^(n - 2))/(factorial(n - 1))*(x)^(n), n = 1..infinity) ProductLog[0, x]= Sum[(- 1)^(n - 1)*Divide[(n)^(n - 2),(n - 1)!]*(x)^(n), {n, 1, Infinity}] Failure Successful Skip -
4.13.E6 W ⁑ ( - e - 1 - ( t 2 / 2 ) ) = βˆ‘ n = 0 ∞ ( - 1 ) n - 1 ⁒ c n ⁒ t n Lambert-W superscript 𝑒 1 superscript 𝑑 2 2 superscript subscript 𝑛 0 superscript 1 𝑛 1 subscript 𝑐 𝑛 superscript 𝑑 𝑛 {\displaystyle{\displaystyle W\left(-e^{-1-(t^{2}/2)}\right)=\sum_{n=0}^{% \infty}(-1)^{n-1}c_{n}t^{n}}} LambertW(- exp(- 1 -((t)^(2)/ 2)))= sum((- 1)^(n - 1)* c[n]*(t)^(n), n = 0..infinity) ProductLog[- Exp[- 1 -((t)^(2)/ 2)]]= Sum[(- 1)^(n - 1)* Subscript[c, n]*(t)^(n), {n, 0, Infinity}] Failure Failure Skip Successful
4.14.E1 sin ⁑ z = e i ⁒ z - e - i ⁒ z 2 ⁒ i 𝑧 superscript 𝑒 imaginary-unit 𝑧 superscript 𝑒 imaginary-unit 𝑧 2 imaginary-unit {\displaystyle{\displaystyle\sin z=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2% \mathrm{i}}}} sin(z)=(exp(I*z)- exp(- I*z))/(2*I) Sin[z]=Divide[Exp[I*z]- Exp[- I*z],2*I] Successful Successful - -
4.14.E2 cos ⁑ z = e i ⁒ z + e - i ⁒ z 2 𝑧 superscript 𝑒 imaginary-unit 𝑧 superscript 𝑒 imaginary-unit 𝑧 2 {\displaystyle{\displaystyle\cos z=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2}}} cos(z)=(exp(I*z)+ exp(- I*z))/(2) Cos[z]=Divide[Exp[I*z]+ Exp[- I*z],2] Successful Successful - -
4.14.E3 cos ⁑ z + i ⁒ sin ⁑ z = e + i ⁒ z 𝑧 𝑖 𝑧 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle\cos z+i\sin z=e^{+iz}}} cos(z)+ I*sin(z)= exp(+ I*z) Cos[z]+ I*Sin[z]= Exp[+ I*z] Successful Successful - -
4.14.E3 cos ⁑ z - i ⁒ sin ⁑ z = e - i ⁒ z 𝑧 𝑖 𝑧 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle\cos z-i\sin z=e^{-iz}}} cos(z)- I*sin(z)= exp(- I*z) Cos[z]- I*Sin[z]= Exp[- I*z] Successful Successful - -
4.14.E4 tan ⁑ z = sin ⁑ z cos ⁑ z 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\tan z=\frac{\sin z}{\cos z}}} tan(z)=(sin(z))/(cos(z)) Tan[z]=Divide[Sin[z],Cos[z]] Successful Successful - -
4.14.E5 csc ⁑ z = 1 sin ⁑ z 𝑧 1 𝑧 {\displaystyle{\displaystyle\csc z=\frac{1}{\sin z}}} csc(z)=(1)/(sin(z)) Csc[z]=Divide[1,Sin[z]] Successful Successful - -
4.14.E6 sec ⁑ z = 1 cos ⁑ z 𝑧 1 𝑧 {\displaystyle{\displaystyle\sec z=\frac{1}{\cos z}}} sec(z)=(1)/(cos(z)) Sec[z]=Divide[1,Cos[z]] Successful Successful - -
4.14.E7 cot ⁑ z = cos ⁑ z sin ⁑ z 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\cot z=\frac{\cos z}{\sin z}}} cot(z)=(cos(z))/(sin(z)) Cot[z]=Divide[Cos[z],Sin[z]] Successful Successful - -
4.14.E7 cos ⁑ z sin ⁑ z = 1 tan ⁑ z 𝑧 𝑧 1 𝑧 {\displaystyle{\displaystyle\frac{\cos z}{\sin z}=\frac{1}{\tan z}}} (cos(z))/(sin(z))=(1)/(tan(z)) Divide[Cos[z],Sin[z]]=Divide[1,Tan[z]] Successful Successful - -
4.14.E8 sin ⁑ ( z + 2 ⁒ k ⁒ Ο€ ) = sin ⁑ z 𝑧 2 π‘˜ πœ‹ 𝑧 {\displaystyle{\displaystyle\sin\left(z+2k\pi\right)=\sin z}} sin(z + 2*k*Pi)= sin(z) Sin[z + 2*k*Pi]= Sin[z] Failure Failure Successful Successful
4.14.E9 cos ⁑ ( z + 2 ⁒ k ⁒ Ο€ ) = cos ⁑ z 𝑧 2 π‘˜ πœ‹ 𝑧 {\displaystyle{\displaystyle\cos\left(z+2k\pi\right)=\cos z}} cos(z + 2*k*Pi)= cos(z) Cos[z + 2*k*Pi]= Cos[z] Failure Failure Successful Successful
4.14.E10 tan ⁑ ( z + k ⁒ Ο€ ) = tan ⁑ z 𝑧 π‘˜ πœ‹ 𝑧 {\displaystyle{\displaystyle\tan\left(z+k\pi\right)=\tan z}} tan(z + k*Pi)= tan(z) Tan[z + k*Pi]= Tan[z] Failure Failure Successful Successful
4.15.E1 cos ⁑ ( x + i ⁒ y ) = sin ⁑ ( x + 1 2 ⁒ Ο€ + i ⁒ y ) π‘₯ 𝑖 𝑦 π‘₯ 1 2 πœ‹ 𝑖 𝑦 {\displaystyle{\displaystyle\cos\left(x+iy\right)=\sin\left(x+\tfrac{1}{2}\pi+% iy\right)}} cos(x + I*y)= sin(x +(1)/(2)*Pi + I*y) Cos[x + I*y]= Sin[x +Divide[1,2]*Pi + I*y] Successful Successful - -
4.15.E2 cot ⁑ ( x + i ⁒ y ) = - tan ⁑ ( x + 1 2 ⁒ Ο€ + i ⁒ y ) π‘₯ 𝑖 𝑦 π‘₯ 1 2 πœ‹ 𝑖 𝑦 {\displaystyle{\displaystyle\cot\left(x+iy\right)=-\tan\left(x+\tfrac{1}{2}\pi% +iy\right)}} cot(x + I*y)= - tan(x +(1)/(2)*Pi + I*y) Cot[x + I*y]= - Tan[x +Divide[1,2]*Pi + I*y] Successful Successful - -
4.15.E3 sec ⁑ ( x + i ⁒ y ) = csc ⁑ ( x + 1 2 ⁒ Ο€ + i ⁒ y ) π‘₯ 𝑖 𝑦 π‘₯ 1 2 πœ‹ 𝑖 𝑦 {\displaystyle{\displaystyle\sec\left(x+iy\right)=\csc\left(x+\tfrac{1}{2}\pi+% iy\right)}} sec(x + I*y)= csc(x +(1)/(2)*Pi + I*y) Sec[x + I*y]= Csc[x +Divide[1,2]*Pi + I*y] Successful Successful - -
4.17.E1 lim z β†’ 0 ⁑ sin ⁑ z z = 1 subscript β†’ 𝑧 0 𝑧 𝑧 1 {\displaystyle{\displaystyle\lim_{z\to 0}\frac{\sin z}{z}=1}} limit((sin(z))/(z), z = 0)= 1 Limit[Divide[Sin[z],z], z -> 0]= 1 Successful Successful - -
4.17.E2 lim z β†’ 0 ⁑ tan ⁑ z z = 1 subscript β†’ 𝑧 0 𝑧 𝑧 1 {\displaystyle{\displaystyle\lim_{z\to 0}\frac{\tan z}{z}=1}} limit((tan(z))/(z), z = 0)= 1 Limit[Divide[Tan[z],z], z -> 0]= 1 Successful Successful - -
4.17.E3 lim z β†’ 0 ⁑ 1 - cos ⁑ z z 2 = 1 2 subscript β†’ 𝑧 0 1 𝑧 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}}} limit((1 - cos(z))/((z)^(2)), z = 0)=(1)/(2) Limit[Divide[1 - Cos[z],(z)^(2)], z -> 0]=Divide[1,2] Successful Successful - -
4.18.E1 2 ⁒ x Ο€ ≀ sin ⁑ x 2 π‘₯ πœ‹ π‘₯ {\displaystyle{\displaystyle\frac{2x}{\pi}<=\sin x}} (2*x)/(Pi)< = sin(x) Divide[2*x,Pi]< = Sin[x] Failure Failure Skip Successful
4.18.E1 sin ⁑ x ≀ x π‘₯ π‘₯ {\displaystyle{\displaystyle\sin x<=x}} sin(x)< = x Sin[x]< = x Failure Failure Skip Successful
4.18.E2 x ≀ tan ⁑ x π‘₯ π‘₯ {\displaystyle{\displaystyle x<=\tan x}} x < = tan(x) x < = Tan[x] Failure Failure Skip Successful
4.18.E3 cos ⁑ x ≀ sin ⁑ x x π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\cos x<=\frac{\sin x}{x}}} cos(x)< =(sin(x))/(x) Cos[x]< =Divide[Sin[x],x] Failure Failure Skip Successful
4.18.E3 sin ⁑ x x ≀ 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle\frac{\sin x}{x}<=1}} (sin(x))/(x)< = 1 Divide[Sin[x],x]< = 1 Failure Failure Skip Successful
4.18.E4 Ο€ < sin ⁑ ( Ο€ ⁒ x ) x ⁒ ( 1 - x ) πœ‹ πœ‹ π‘₯ π‘₯ 1 π‘₯ {\displaystyle{\displaystyle\pi<\frac{\sin\left(\pi x\right)}{x(1-x)}}} Pi <(sin(Pi*x))/(x*(1 - x)) Pi <Divide[Sin[Pi*x],x*(1 - x)] Failure Failure Successful Successful
4.18.E4 sin ⁑ ( Ο€ ⁒ x ) x ⁒ ( 1 - x ) ≀ 4 πœ‹ π‘₯ π‘₯ 1 π‘₯ 4 {\displaystyle{\displaystyle\frac{\sin\left(\pi x\right)}{x(1-x)}<=4}} (sin(Pi*x))/(x*(1 - x))< = 4 Divide[Sin[Pi*x],x*(1 - x)]< = 4 Failure Failure Successful Successful
4.18.E5 | sinh ⁑ y | ≀ | sin ⁑ z | ≀ cosh ⁑ y 𝑦 𝑧 𝑦 {\displaystyle{\displaystyle|\sinh y|<=|\sin z|\leq\cosh y}} abs(sinh(y))< =abs(sin(z))<= cosh(y) Abs[Sinh[y]]< =Abs[Sin[z]]<= Cosh[y] Failure Failure Error Successful
4.18.E6 | sinh ⁑ y | ≀ | cos ⁑ z | ≀ cosh ⁑ y 𝑦 𝑧 𝑦 {\displaystyle{\displaystyle|\sinh y|<=|\cos z|\leq\cosh y}} abs(sinh(y))< =abs(cos(z))<= cosh(y) Abs[Sinh[y]]< =Abs[Cos[z]]<= Cosh[y] Failure Failure Error Successful
4.18.E7 | csc ⁑ z | ≀ csch ⁑ | y | 𝑧 𝑦 {\displaystyle{\displaystyle|\csc z|<=\operatorname{csch}|y|}} abs(csc(z))< = csch(abs(y)) Abs[Csc[z]]< = Csch[Abs[y]] Failure Failure
Fail
.4602792559 <= .2757205648 <- {z = 2^(1/2)+I*2^(1/2), y = 2}
.4602792559 <= .9982156967e-1 <- {z = 2^(1/2)+I*2^(1/2), y = 3}
.4602792559 <= .2757205648 <- {z = 2^(1/2)-I*2^(1/2), y = 2}
.4602792559 <= .9982156967e-1 <- {z = 2^(1/2)-I*2^(1/2), y = 3}
... skip entries to safe data
 Successful
4.18.E8 | cos ⁑ z | ≀ cosh ⁑ | z | 𝑧 𝑧 {\displaystyle{\displaystyle|\cos z|<=\cosh|z|}} abs(cos(z))< = cosh(abs(z)) Abs[Cos[z]]< = Cosh[Abs[z]] Failure Failure Successful Successful
4.18.E9 | sin ⁑ z | ≀ sinh ⁑ | z | 𝑧 𝑧 {\displaystyle{\displaystyle|\sin z|<=\sinh|z|}} abs(sin(z))< = sinh(abs(z)) Abs[Sin[z]]< = Sinh[Abs[z]] Failure Failure Successful Successful
4.18#Ex1 | cos ⁑ z | < 2 𝑧 2 {\displaystyle{\displaystyle|\cos z|<2}} abs(cos(z))< 2 Abs[Cos[z]]< 2 Failure Failure Successful Successful
4.18#Ex2 | sin ⁑ z | ≀ 6 5 ⁒ | z | 𝑧 6 5 𝑧 {\displaystyle{\displaystyle|\sin z|<=\tfrac{6}{5}|z|}} abs(sin(z))< =(6)/(5)*abs(z) Abs[Sin[z]]< =Divide[6,5]*Abs[z] Failure Failure Successful Successful
4.19.E7 ln ⁑ ( sin ⁑ z z ) = βˆ‘ n = 1 ∞ ( - 1 ) n ⁒ 2 2 ⁒ n - 1 ⁒ B 2 ⁒ n n ⁒ ( 2 ⁒ n ) ! ⁒ z 2 ⁒ n 𝑧 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 2 2 𝑛 1 Bernoulli-number-B 2 𝑛 𝑛 2 𝑛 superscript 𝑧 2 𝑛 {\displaystyle{\displaystyle\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n}2^{2n-1}B_{2n}}{n(2n)!}z^{2n}}} ln((sin(z))/(z))= sum(((- 1)^(n)* (2)^(2*n - 1)* bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity) Log[Divide[Sin[z],z]]= Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)* BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}] Failure Failure Skip Successful
4.19.E8 ln ⁑ ( cos ⁑ z ) = βˆ‘ n = 1 ∞ ( - 1 ) n ⁒ 2 2 ⁒ n - 1 ⁒ ( 2 2 ⁒ n - 1 ) ⁒ B 2 ⁒ n n ⁒ ( 2 ⁒ n ) ! ⁒ z 2 ⁒ n 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 2 2 𝑛 1 superscript 2 2 𝑛 1 Bernoulli-number-B 2 𝑛 𝑛 2 𝑛 superscript 𝑧 2 𝑛 {\displaystyle{\displaystyle\ln\left(\cos z\right)=\sum_{n=1}^{\infty}\frac{(-% 1)^{n}2^{2n-1}(2^{2n}-1)B_{2n}}{n(2n)!}z^{2n}}} ln(cos(z))= sum(((- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)* bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity) Log[Cos[z]]= Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)* BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}] Failure Failure Skip Successful
4.19.E9 ln ⁑ ( tan ⁑ z z ) = βˆ‘ n = 1 ∞ ( - 1 ) n - 1 ⁒ 2 2 ⁒ n ⁒ ( 2 2 ⁒ n - 1 - 1 ) ⁒ B 2 ⁒ n n ⁒ ( 2 ⁒ n ) ! ⁒ z 2 ⁒ n 𝑧 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 1 superscript 2 2 𝑛 superscript 2 2 𝑛 1 1 Bernoulli-number-B 2 𝑛 𝑛 2 𝑛 superscript 𝑧 2 𝑛 {\displaystyle{\displaystyle\ln\left(\frac{\tan z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}z^{2n}}} ln((tan(z))/(z))= sum(((- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)* bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity) Log[Divide[Tan[z],z]]= Sum[Divide[(- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)* BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}] Failure Failure Skip Successful
4.20.E1 d d z ⁑ sin ⁑ z = cos ⁑ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sin z=\cos z}} diff(sin(z), z)= cos(z) D[Sin[z], z]= Cos[z] Successful Successful - -
4.20.E2 d d z ⁑ cos ⁑ z = - sin ⁑ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cos z=-\sin z}} diff(cos(z), z)= - sin(z) D[Cos[z], z]= - Sin[z] Successful Successful - -
4.20.E3 d d z ⁑ tan ⁑ z = sec 2 ⁑ z derivative 𝑧 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tan z={\sec^{2}}z}} diff(tan(z), z)= (sec(z))^(2) D[Tan[z], z]= (Sec[z])^(2) Successful Successful - -
4.20.E4 d d z ⁑ csc ⁑ z = - csc ⁑ z ⁒ cot ⁑ z derivative 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\csc z=-\csc z\cot z}} diff(csc(z), z)= - csc(z)*cot(z) D[Csc[z], z]= - Csc[z]*Cot[z] Successful Successful - -
4.20.E5 d d z ⁑ sec ⁑ z = sec ⁑ z ⁒ tan ⁑ z derivative 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sec z=\sec z\tan z}} diff(sec(z), z)= sec(z)*tan(z) D[Sec[z], z]= Sec[z]*Tan[z] Successful Successful - -
4.20.E6 d d z ⁑ cot ⁑ z = - csc 2 ⁑ z derivative 𝑧 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cot z=-{\csc^{2}}z}} diff(cot(z), z)= - (csc(z))^(2) D[Cot[z], z]= - (Csc[z])^(2) Successful Successful - -
4.20.E7 d n d z n ⁑ sin ⁑ z = sin ⁑ ( z + 1 2 ⁒ n ⁒ Ο€ ) derivative 𝑧 𝑛 𝑧 𝑧 1 2 𝑛 πœ‹ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\sin z=% \sin\left(z+\tfrac{1}{2}n\pi\right)}} diff(sin(z), [z$(n)])= sin(z +(1)/(2)*n*Pi) D[Sin[z], {z, n}]= Sin[z +Divide[1,2]*n*Pi] Successful Successful - -
4.20.E8 d n d z n ⁑ cos ⁑ z = cos ⁑ ( z + 1 2 ⁒ n ⁒ Ο€ ) derivative 𝑧 𝑛 𝑧 𝑧 1 2 𝑛 πœ‹ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cos z=% \cos\left(z+\tfrac{1}{2}n\pi\right)}} diff(cos(z), [z$(n)])= cos(z +(1)/(2)*n*Pi) D[Cos[z], {z, n}]= Cos[z +Divide[1,2]*n*Pi] Successful Successful - -
4.20.E9 d 2 w d z 2 + a 2 ⁒ w = 0 derivative 𝑀 𝑧 2 superscript π‘Ž 2 𝑀 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w% =0}} diff(w, [z$(2)])+ (a)^(2)* w = 0 D[w, {z, 2}]+ (a)^(2)* w = 0 Failure Failure
Fail
-5.656854245+5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
5.656854245+5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
5.656854245-5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-5.656854245-5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.20.E10 ( d w d z ) 2 + a 2 ⁒ w 2 = 1 superscript derivative 𝑀 𝑧 2 superscript π‘Ž 2 superscript 𝑀 2 1 {\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a% ^{2}w^{2}=1}} (diff(w, z))^(2)+ (a)^(2)* (w)^(2)= 1 (D[w, z])^(2)+ (a)^(2)* (w)^(2)= 1 Failure Failure
Fail
-16.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
14.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
-16.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
14.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.20.E11 d w d z - a 2 ⁒ w 2 = 1 derivative 𝑀 𝑧 superscript π‘Ž 2 superscript 𝑀 2 1 {\displaystyle{\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}-a^{2}w^{2}=1}} diff(w, z)- (a)^(2)* (w)^(2)= 1 D[w, z]- (a)^(2)* (w)^(2)= 1 Failure Failure
Fail
14.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
-16.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
14.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-16.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.20.E12 w = A ⁒ cos ⁑ ( a ⁒ z ) + B ⁒ sin ⁑ ( a ⁒ z ) 𝑀 𝐴 π‘Ž 𝑧 𝐡 π‘Ž 𝑧 {\displaystyle{\displaystyle w=A\cos\left(az\right)+B\sin\left(az\right)}} w = A*cos(a*z)+ B*sin(a*z) w = A*Cos[a*z]+ B*Sin[a*z] Failure Failure Skip Skip
4.20.E13 w = ( 1 / a ) ⁒ sin ⁑ ( a ⁒ z + c ) 𝑀 1 π‘Ž π‘Ž 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\sin\left(az+c\right)}} w =(1/ a)* sin(a*z + c) w =(1/ a)* Sin[a*z + c] Failure Failure
Fail
-43.99146068+34.43827298*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.560620374+.384416402*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-.552876601+4.108989171*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
2.401710418+1.589052846*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-43.991460739515965, 34.43827305491785] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.5606203716754656, 0.38441640190707305] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.5528766038746884, 4.1089891749071095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.4017104180648507, 1.5890528479992119] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.20.E14 w = ( 1 / a ) ⁒ tan ⁑ ( a ⁒ z + c ) 𝑀 1 π‘Ž π‘Ž 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\tan\left(az+c\right)}} w =(1/ a)* tan(a*z + c) w =(1/ a)* Tan[a*z + c] Failure Failure
Fail
1.060642513+1.060651152*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.097992560+1.014214371*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.770353735+1.772853405*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.045061232+1.116024928*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.0606425136739976, 1.0606511525471942] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.0979925605963208, 1.0142143722877455] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.7703537351803704, 1.7728534052480869] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.0450612330665354, 1.11602492841073] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E1 sin ⁑ u + cos ⁑ u = 2 ⁒ sin ⁑ ( u + 1 4 ⁒ Ο€ ) 𝑒 𝑒 2 𝑒 1 4 πœ‹ {\displaystyle{\displaystyle\sin u+\cos u=\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi% \right)}} sin(u)+ cos(u)=sqrt(2)*sin(u +(1)/(4)*Pi) Sin[u]+ Cos[u]=Sqrt[2]*Sin[u +Divide[1,4]*Pi] Successful Successful - -
4.21.E1 sin ⁑ u - cos ⁑ u = 2 ⁒ sin ⁑ ( u - 1 4 ⁒ Ο€ ) 𝑒 𝑒 2 𝑒 1 4 πœ‹ {\displaystyle{\displaystyle\sin u-\cos u=\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi% \right)}} sin(u)- cos(u)=sqrt(2)*sin(u -(1)/(4)*Pi) Sin[u]- Cos[u]=Sqrt[2]*Sin[u -Divide[1,4]*Pi] Successful Successful - -
4.21.E1 2 ⁒ sin ⁑ ( u + 1 4 ⁒ Ο€ ) = + 2 ⁒ cos ⁑ ( u - 1 4 ⁒ Ο€ ) 2 𝑒 1 4 πœ‹ 2 𝑒 1 4 πœ‹ {\displaystyle{\displaystyle\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi\right)=+\sqrt{% 2}\cos\left(u-\tfrac{1}{4}\pi\right)}} sqrt(2)*sin(u +(1)/(4)*Pi)= +sqrt(2)*cos(u -(1)/(4)*Pi) Sqrt[2]*Sin[u +Divide[1,4]*Pi]= +Sqrt[2]*Cos[u -Divide[1,4]*Pi] Successful Successful - -
4.21.E1 2 ⁒ sin ⁑ ( u - 1 4 ⁒ Ο€ ) = - 2 ⁒ cos ⁑ ( u + 1 4 ⁒ Ο€ ) 2 𝑒 1 4 πœ‹ 2 𝑒 1 4 πœ‹ {\displaystyle{\displaystyle\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi\right)=-\sqrt{% 2}\cos\left(u+\tfrac{1}{4}\pi\right)}} sqrt(2)*sin(u -(1)/(4)*Pi)= -sqrt(2)*cos(u +(1)/(4)*Pi) Sqrt[2]*Sin[u -Divide[1,4]*Pi]= -Sqrt[2]*Cos[u +Divide[1,4]*Pi] Successful Successful - -
4.21.E2 sin ⁑ ( u + v ) = sin ⁑ u ⁒ cos ⁑ v + cos ⁑ u ⁒ sin ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\sin\left(u+v\right)=\sin u\cos v+\cos u\sin v}} sin(u + v)= sin(u)*cos(v)+ cos(u)*sin(v) Sin[u + v]= Sin[u]*Cos[v]+ Cos[u]*Sin[v] Successful Successful - -
4.21.E2 sin ⁑ ( u - v ) = sin ⁑ u ⁒ cos ⁑ v - cos ⁑ u ⁒ sin ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\sin\left(u-v\right)=\sin u\cos v-\cos u\sin v}} sin(u - v)= sin(u)*cos(v)- cos(u)*sin(v) Sin[u - v]= Sin[u]*Cos[v]- Cos[u]*Sin[v] Successful Successful - -
4.21.E3 cos ⁑ ( u + v ) = cos ⁑ u ⁒ cos ⁑ v - sin ⁑ u ⁒ sin ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\cos\left(u+v\right)=\cos u\cos v-\sin u\sin v}} cos(u + v)= cos(u)*cos(v)- sin(u)*sin(v) Cos[u + v]= Cos[u]*Cos[v]- Sin[u]*Sin[v] Successful Successful - -
4.21.E3 cos ⁑ ( u - v ) = cos ⁑ u ⁒ cos ⁑ v + sin ⁑ u ⁒ sin ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\cos\left(u-v\right)=\cos u\cos v+\sin u\sin v}} cos(u - v)= cos(u)*cos(v)+ sin(u)*sin(v) Cos[u - v]= Cos[u]*Cos[v]+ Sin[u]*Sin[v] Successful Successful - -
4.21.E4 tan ⁑ ( u + v ) = tan ⁑ u + tan ⁑ v 1 - tan ⁑ u ⁒ tan ⁑ v 𝑒 𝑣 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\tan\left(u+v\right)=\frac{\tan u+\tan v}{1-\tan u% \tan v}}} tan(u + v)=(tan(u)+ tan(v))/(1 - tan(u)*tan(v)) Tan[u + v]=Divide[Tan[u]+ Tan[v],1 - Tan[u]*Tan[v]] Successful Successful - -
4.21.E4 tan ⁑ ( u - v ) = tan ⁑ u - tan ⁑ v 1 + tan ⁑ u ⁒ tan ⁑ v 𝑒 𝑣 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\tan\left(u-v\right)=\frac{\tan u-\tan v}{1+\tan u% \tan v}}} tan(u - v)=(tan(u)- tan(v))/(1 + tan(u)*tan(v)) Tan[u - v]=Divide[Tan[u]- Tan[v],1 + Tan[u]*Tan[v]] Successful Successful - -
4.21.E5 cot ⁑ ( u + v ) = + cot ⁑ u ⁒ cot ⁑ v - 1 cot ⁑ u + cot ⁑ v 𝑒 𝑣 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\cot\left(u+v\right)=\frac{+\cot u\cot v-1}{\cot u% +\cot v}}} cot(u + v)=(+ cot(u)*cot(v)- 1)/(cot(u)+ cot(v)) Cot[u + v]=Divide[+ Cot[u]*Cot[v]- 1,Cot[u]+ Cot[v]] Successful Successful - -
4.21.E5 cot ⁑ ( u - v ) = - cot ⁑ u ⁒ cot ⁑ v - 1 cot ⁑ u - cot ⁑ v 𝑒 𝑣 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\cot\left(u-v\right)=\frac{-\cot u\cot v-1}{\cot u% -\cot v}}} cot(u - v)=(- cot(u)*cot(v)- 1)/(cot(u)- cot(v)) Cot[u - v]=Divide[- Cot[u]*Cot[v]- 1,Cot[u]- Cot[v]] Successful Successful - -
4.21.E6 sin ⁑ u + sin ⁑ v = 2 ⁒ sin ⁑ ( u + v 2 ) ⁒ cos ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\sin u+\sin v=2\sin\left(\frac{u+v}{2}\right)\cos% \left(\frac{u-v}{2}\right)}} sin(u)+ sin(v)= 2*sin((u + v)/(2))*cos((u - v)/(2)) Sin[u]+ Sin[v]= 2*Sin[Divide[u + v,2]]*Cos[Divide[u - v,2]] Successful Successful - -
4.21.E7 sin ⁑ u - sin ⁑ v = 2 ⁒ cos ⁑ ( u + v 2 ) ⁒ sin ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\sin u-\sin v=2\cos\left(\frac{u+v}{2}\right)\sin% \left(\frac{u-v}{2}\right)}} sin(u)- sin(v)= 2*cos((u + v)/(2))*sin((u - v)/(2)) Sin[u]- Sin[v]= 2*Cos[Divide[u + v,2]]*Sin[Divide[u - v,2]] Successful Successful - -
4.21.E8 cos ⁑ u + cos ⁑ v = 2 ⁒ cos ⁑ ( u + v 2 ) ⁒ cos ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\cos u+\cos v=2\cos\left(\frac{u+v}{2}\right)\cos% \left(\frac{u-v}{2}\right)}} cos(u)+ cos(v)= 2*cos((u + v)/(2))*cos((u - v)/(2)) Cos[u]+ Cos[v]= 2*Cos[Divide[u + v,2]]*Cos[Divide[u - v,2]] Successful Successful - -
4.21.E9 cos ⁑ u - cos ⁑ v = - 2 ⁒ sin ⁑ ( u + v 2 ) ⁒ sin ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\cos u-\cos v=-2\sin\left(\frac{u+v}{2}\right)\sin% \left(\frac{u-v}{2}\right)}} cos(u)- cos(v)= - 2*sin((u + v)/(2))*sin((u - v)/(2)) Cos[u]- Cos[v]= - 2*Sin[Divide[u + v,2]]*Sin[Divide[u - v,2]] Successful Successful - -
4.21.E10 tan ⁑ u + tan ⁑ v = sin ⁑ ( u + v ) cos ⁑ u ⁒ cos ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\tan u+\tan v=\frac{\sin\left(u+v\right)}{\cos u% \cos v}}} tan(u)+ tan(v)=(sin(u + v))/(cos(u)*cos(v)) Tan[u]+ Tan[v]=Divide[Sin[u + v],Cos[u]*Cos[v]] Successful Successful - -
4.21.E10 tan ⁑ u - tan ⁑ v = sin ⁑ ( u - v ) cos ⁑ u ⁒ cos ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\tan u-\tan v=\frac{\sin\left(u-v\right)}{\cos u% \cos v}}} tan(u)- tan(v)=(sin(u - v))/(cos(u)*cos(v)) Tan[u]- Tan[v]=Divide[Sin[u - v],Cos[u]*Cos[v]] Successful Successful - -
4.21.E11 cot ⁑ u + cot ⁑ v = sin ⁑ ( v + u ) sin ⁑ u ⁒ sin ⁑ v 𝑒 𝑣 𝑣 𝑒 𝑒 𝑣 {\displaystyle{\displaystyle\cot u+\cot v=\frac{\sin\left(v+u\right)}{\sin u% \sin v}}} cot(u)+ cot(v)=(sin(v + u))/(sin(u)*sin(v)) Cot[u]+ Cot[v]=Divide[Sin[v + u],Sin[u]*Sin[v]] Successful Successful - -
4.21.E11 cot ⁑ u - cot ⁑ v = sin ⁑ ( v - u ) sin ⁑ u ⁒ sin ⁑ v 𝑒 𝑣 𝑣 𝑒 𝑒 𝑣 {\displaystyle{\displaystyle\cot u-\cot v=\frac{\sin\left(v-u\right)}{\sin u% \sin v}}} cot(u)- cot(v)=(sin(v - u))/(sin(u)*sin(v)) Cot[u]- Cot[v]=Divide[Sin[v - u],Sin[u]*Sin[v]] Successful Successful - -
4.21.E12 sin 2 ⁑ z + cos 2 ⁑ z = 1 2 𝑧 2 𝑧 1 {\displaystyle{\displaystyle{\sin^{2}}z+{\cos^{2}}z=1}} (sin(z))^(2)+ (cos(z))^(2)= 1 (Sin[z])^(2)+ (Cos[z])^(2)= 1 Successful Successful - -
4.21.E13 sec 2 ⁑ z = 1 + tan 2 ⁑ z 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle{\sec^{2}}z=1+{\tan^{2}}z}} (sec(z))^(2)= 1 + (tan(z))^(2) (Sec[z])^(2)= 1 + (Tan[z])^(2) Successful Successful - -
4.21.E14 csc 2 ⁑ z = 1 + cot 2 ⁑ z 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle{\csc^{2}}z=1+{\cot^{2}}z}} (csc(z))^(2)= 1 + (cot(z))^(2) (Csc[z])^(2)= 1 + (Cot[z])^(2) Successful Successful - -
4.21.E15 2 ⁒ sin ⁑ u ⁒ sin ⁑ v = cos ⁑ ( u - v ) - cos ⁑ ( u + v ) 2 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle 2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v% \right)}} 2*sin(u)*sin(v)= cos(u - v)- cos(u + v) 2*Sin[u]*Sin[v]= Cos[u - v]- Cos[u + v] Successful Successful - -
4.21.E16 2 ⁒ cos ⁑ u ⁒ cos ⁑ v = cos ⁑ ( u - v ) + cos ⁑ ( u + v ) 2 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle 2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v% \right)}} 2*cos(u)*cos(v)= cos(u - v)+ cos(u + v) 2*Cos[u]*Cos[v]= Cos[u - v]+ Cos[u + v] Successful Successful - -
4.21.E17 2 ⁒ sin ⁑ u ⁒ cos ⁑ v = sin ⁑ ( u - v ) + sin ⁑ ( u + v ) 2 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle 2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v% \right)}} 2*sin(u)*cos(v)= sin(u - v)+ sin(u + v) 2*Sin[u]*Cos[v]= Sin[u - v]+ Sin[u + v] Successful Successful - -
4.21.E18 sin 2 ⁑ u - sin 2 ⁑ v = sin ⁑ ( u + v ) ⁒ sin ⁑ ( u - v ) 2 𝑒 2 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle{\sin^{2}}u-{\sin^{2}}v=\sin\left(u+v\right)\sin% \left(u-v\right)}} (sin(u))^(2)- (sin(v))^(2)= sin(u + v)*sin(u - v) (Sin[u])^(2)- (Sin[v])^(2)= Sin[u + v]*Sin[u - v] Successful Successful - -
4.21.E19 cos 2 ⁑ u - cos 2 ⁑ v = - sin ⁑ ( u + v ) ⁒ sin ⁑ ( u - v ) 2 𝑒 2 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle{\cos^{2}}u-{\cos^{2}}v=-\sin\left(u+v\right)\sin% \left(u-v\right)}} (cos(u))^(2)- (cos(v))^(2)= - sin(u + v)*sin(u - v) (Cos[u])^(2)- (Cos[v])^(2)= - Sin[u + v]*Sin[u - v] Successful Successful - -
4.21.E20 cos 2 ⁑ u - sin 2 ⁑ v = cos ⁑ ( u + v ) ⁒ cos ⁑ ( u - v ) 2 𝑒 2 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle{\cos^{2}}u-{\sin^{2}}v=\cos\left(u+v\right)\cos% \left(u-v\right)}} (cos(u))^(2)- (sin(v))^(2)= cos(u + v)*cos(u - v) (Cos[u])^(2)- (Sin[v])^(2)= Cos[u + v]*Cos[u - v] Successful Successful - -
4.21.E21 sin ⁑ z 2 = + ( 1 - cos ⁑ z 2 ) 1 / 2 𝑧 2 superscript 1 𝑧 2 1 2 {\displaystyle{\displaystyle\sin\frac{z}{2}=+\left(\frac{1-\cos z}{2}\right)^{% 1/2}}} sin((z)/(2))= +((1 - cos(z))/(2))^(1/ 2) Sin[Divide[z,2]]= +(Divide[1 - Cos[z],2])^(1/ 2) Failure Failure
Fail
-1.637854044-1.167010648*I <- {z = -2^(1/2)-I*2^(1/2)}
-1.637854044+1.167010648*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-1.6378540442140963, -1.1670106484252494] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6378540442140963, 1.1670106484252494] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.21.E21 sin ⁑ z 2 = - ( 1 - cos ⁑ z 2 ) 1 / 2 𝑧 2 superscript 1 𝑧 2 1 2 {\displaystyle{\displaystyle\sin\frac{z}{2}=-\left(\frac{1-\cos z}{2}\right)^{% 1/2}}} sin((z)/(2))= -((1 - cos(z))/(2))^(1/ 2) Sin[Divide[z,2]]= -(Divide[1 - Cos[z],2])^(1/ 2) Failure Failure
Fail
1.637854044+1.167010648*I <- {z = 2^(1/2)+I*2^(1/2)}
1.637854044-1.167010648*I <- {z = 2^(1/2)-I*2^(1/2)}
Fail
Complex[1.6378540442140963, 1.1670106484252494] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.6378540442140963, -1.1670106484252494] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
4.21.E22 cos ⁑ z 2 = + ( 1 + cos ⁑ z 2 ) 1 / 2 𝑧 2 superscript 1 𝑧 2 1 2 {\displaystyle{\displaystyle\cos\frac{z}{2}=+\left(\frac{1+\cos z}{2}\right)^{% 1/2}}} cos((z)/(2))= +((1 + cos(z))/(2))^(1/ 2) Cos[Divide[z,2]]= +(Divide[1 + Cos[z],2])^(1/ 2) Failure Failure Successful Successful
4.21.E22 cos ⁑ z 2 = - ( 1 + cos ⁑ z 2 ) 1 / 2 𝑧 2 superscript 1 𝑧 2 1 2 {\displaystyle{\displaystyle\cos\frac{z}{2}=-\left(\frac{1+\cos z}{2}\right)^{% 1/2}}} cos((z)/(2))= -((1 + cos(z))/(2))^(1/ 2) Cos[Divide[z,2]]= -(Divide[1 + Cos[z],2])^(1/ 2) Failure Failure
Fail
1.916716266-.9972227728*I <- {z = 2^(1/2)+I*2^(1/2)}
1.916716266+.9972227728*I <- {z = 2^(1/2)-I*2^(1/2)}
1.916716266-.9972227728*I <- {z = -2^(1/2)-I*2^(1/2)}
1.916716266+.9972227728*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.916716265666014, -0.9972227733456656] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.916716265666014, 0.9972227733456656] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.916716265666014, -0.9972227733456656] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.916716265666014, 0.9972227733456656] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.21.E23 tan ⁑ z 2 = + ( 1 - cos ⁑ z 1 + cos ⁑ z ) 1 / 2 𝑧 2 superscript 1 𝑧 1 𝑧 1 2 {\displaystyle{\displaystyle\tan\frac{z}{2}=+\left(\frac{1-\cos z}{1+\cos z}% \right)^{1/2}}} tan((z)/(2))= +((1 - cos(z))/(1 + cos(z)))^(1/ 2) Tan[Divide[z,2]]= +(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2) Failure Failure
Fail
-.8463685478-1.658064547*I <- {z = -2^(1/2)-I*2^(1/2)}
-.8463685478+1.658064547*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.8463685477398917, -1.6580645472823399] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.8463685477398917, 1.6580645472823399] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.21.E23 tan ⁑ z 2 = - ( 1 - cos ⁑ z 1 + cos ⁑ z ) 1 / 2 𝑧 2 superscript 1 𝑧 1 𝑧 1 2 {\displaystyle{\displaystyle\tan\frac{z}{2}=-\left(\frac{1-\cos z}{1+\cos z}% \right)^{1/2}}} tan((z)/(2))= -((1 - cos(z))/(1 + cos(z)))^(1/ 2) Tan[Divide[z,2]]= -(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2) Failure Failure
Fail
.8463685478+1.658064547*I <- {z = 2^(1/2)+I*2^(1/2)}
.8463685478-1.658064547*I <- {z = 2^(1/2)-I*2^(1/2)}
Fail
Complex[0.8463685477398917, 1.6580645472823399] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8463685477398917, -1.6580645472823399] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
4.21.E23 + ( 1 - cos ⁑ z 1 + cos ⁑ z ) 1 / 2 = 1 - cos ⁑ z sin ⁑ z superscript 1 𝑧 1 𝑧 1 2 1 𝑧 𝑧 {\displaystyle{\displaystyle+\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=% \frac{1-\cos z}{\sin z}}} +((1 - cos(z))/(1 + cos(z)))^(1/ 2)=(1 - cos(z))/(sin(z)) +(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2)=Divide[1 - Cos[z],Sin[z]] Failure Failure Skip
Fail
Complex[0.8463685477398916, 1.6580645472823403] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.8463685477398916, -1.6580645472823403] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.21.E23 - ( 1 - cos ⁑ z 1 + cos ⁑ z ) 1 / 2 = 1 - cos ⁑ z sin ⁑ z superscript 1 𝑧 1 𝑧 1 2 1 𝑧 𝑧 {\displaystyle{\displaystyle-\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=% \frac{1-\cos z}{\sin z}}} -((1 - cos(z))/(1 + cos(z)))^(1/ 2)=(1 - cos(z))/(sin(z)) -(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2)=Divide[1 - Cos[z],Sin[z]] Failure Failure Skip
Fail
Complex[-0.8463685477398916, -1.6580645472823403] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.8463685477398916, 1.6580645472823403] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
4.21.E23 1 - cos ⁑ z sin ⁑ z = sin ⁑ z 1 + cos ⁑ z 1 𝑧 𝑧 𝑧 1 𝑧 {\displaystyle{\displaystyle\frac{1-\cos z}{\sin z}=\frac{\sin z}{1+\cos z}}} (1 - cos(z))/(sin(z))=(sin(z))/(1 + cos(z)) Divide[1 - Cos[z],Sin[z]]=Divide[Sin[z],1 + Cos[z]] Successful Successful - -
4.21.E24 sin ⁑ ( - z ) = - sin ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\sin\left(-z\right)=-\sin z}} sin(- z)= - sin(z) Sin[- z]= - Sin[z] Successful Successful - -
4.21.E25 cos ⁑ ( - z ) = cos ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\cos\left(-z\right)=\cos z}} cos(- z)= cos(z) Cos[- z]= Cos[z] Successful Successful - -
4.21.E26 tan ⁑ ( - z ) = - tan ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\tan\left(-z\right)=-\tan z}} tan(- z)= - tan(z) Tan[- z]= - Tan[z] Successful Successful - -
4.21.E27 sin ⁑ ( 2 ⁒ z ) = 2 ⁒ sin ⁑ z ⁒ cos ⁑ z 2 𝑧 2 𝑧 𝑧 {\displaystyle{\displaystyle\sin\left(2z\right)=2\sin z\cos z}} sin(2*z)= 2*sin(z)*cos(z) Sin[2*z]= 2*Sin[z]*Cos[z] Successful Successful - -
4.21.E27 2 ⁒ sin ⁑ z ⁒ cos ⁑ z = 2 ⁒ tan ⁑ z 1 + tan 2 ⁑ z 2 𝑧 𝑧 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle 2\sin z\cos z=\frac{2\tan z}{1+{\tan^{2}}z}}} 2*sin(z)*cos(z)=(2*tan(z))/(1 + (tan(z))^(2)) 2*Sin[z]*Cos[z]=Divide[2*Tan[z],1 + (Tan[z])^(2)] Successful Successful - -
4.21.E28 cos ⁑ ( 2 ⁒ z ) = 2 ⁒ cos 2 ⁑ z - 1 2 𝑧 2 2 𝑧 1 {\displaystyle{\displaystyle\cos\left(2z\right)=2{\cos^{2}}z-1}} cos(2*z)= 2*(cos(z))^(2)- 1 Cos[2*z]= 2*(Cos[z])^(2)- 1 Successful Successful - -
4.21.E28 2 ⁒ cos 2 ⁑ z - 1 = 1 - 2 ⁒ sin 2 ⁑ z 2 2 𝑧 1 1 2 2 𝑧 {\displaystyle{\displaystyle 2{\cos^{2}}z-1=1-2{\sin^{2}}z}} 2*(cos(z))^(2)- 1 = 1 - 2*(sin(z))^(2) 2*(Cos[z])^(2)- 1 = 1 - 2*(Sin[z])^(2) Successful Successful - -
4.21.E28 1 - 2 ⁒ sin 2 ⁑ z = cos 2 ⁑ z - sin 2 ⁑ z 1 2 2 𝑧 2 𝑧 2 𝑧 {\displaystyle{\displaystyle 1-2{\sin^{2}}z={\cos^{2}}z-{\sin^{2}}z}} 1 - 2*(sin(z))^(2)= (cos(z))^(2)- (sin(z))^(2) 1 - 2*(Sin[z])^(2)= (Cos[z])^(2)- (Sin[z])^(2) Successful Successful - -
4.21.E28 cos 2 ⁑ z - sin 2 ⁑ z = 1 - tan 2 ⁑ z 1 + tan 2 ⁑ z 2 𝑧 2 𝑧 1 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle{\cos^{2}}z-{\sin^{2}}z=\frac{1-{\tan^{2}}z}{1+{% \tan^{2}}z}}} (cos(z))^(2)- (sin(z))^(2)=(1 - (tan(z))^(2))/(1 + (tan(z))^(2)) (Cos[z])^(2)- (Sin[z])^(2)=Divide[1 - (Tan[z])^(2),1 + (Tan[z])^(2)] Successful Successful - -
4.21.E29 tan ⁑ ( 2 ⁒ z ) = 2 ⁒ tan ⁑ z 1 - tan 2 ⁑ z 2 𝑧 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle\tan\left(2z\right)=\frac{2\tan z}{1-{\tan^{2}}z}}} tan(2*z)=(2*tan(z))/(1 - (tan(z))^(2)) Tan[2*z]=Divide[2*Tan[z],1 - (Tan[z])^(2)] Successful Successful - -
4.21.E29 2 ⁒ tan ⁑ z 1 - tan 2 ⁑ z = 2 ⁒ cot ⁑ z cot 2 ⁑ z - 1 2 𝑧 1 2 𝑧 2 𝑧 2 𝑧 1 {\displaystyle{\displaystyle\frac{2\tan z}{1-{\tan^{2}}z}=\frac{2\cot z}{{\cot% ^{2}}z-1}}} (2*tan(z))/(1 - (tan(z))^(2))=(2*cot(z))/((cot(z))^(2)- 1) Divide[2*Tan[z],1 - (Tan[z])^(2)]=Divide[2*Cot[z],(Cot[z])^(2)- 1] Successful Successful - -
4.21.E29 2 ⁒ cot ⁑ z cot 2 ⁑ z - 1 = 2 cot ⁑ z - tan ⁑ z 2 𝑧 2 𝑧 1 2 𝑧 𝑧 {\displaystyle{\displaystyle\frac{2\cot z}{{\cot^{2}}z-1}=\frac{2}{\cot z-\tan z% }}} (2*cot(z))/((cot(z))^(2)- 1)=(2)/(cot(z)- tan(z)) Divide[2*Cot[z],(Cot[z])^(2)- 1]=Divide[2,Cot[z]- Tan[z]] Successful Successful - -
4.21.E30 sin ⁑ ( 3 ⁒ z ) = 3 ⁒ sin ⁑ z - 4 ⁒ sin 3 ⁑ z 3 𝑧 3 𝑧 4 3 𝑧 {\displaystyle{\displaystyle\sin\left(3z\right)=3\sin z-4{\sin^{3}}z}} sin(3*z)= 3*sin(z)- 4*(sin(z))^(3) Sin[3*z]= 3*Sin[z]- 4*(Sin[z])^(3) Successful Successful - -
4.21.E31 cos ⁑ ( 3 ⁒ z ) = - 3 ⁒ cos ⁑ z + 4 ⁒ cos 3 ⁑ z 3 𝑧 3 𝑧 4 3 𝑧 {\displaystyle{\displaystyle\cos\left(3z\right)=-3\cos z+4{\cos^{3}}z}} cos(3*z)= - 3*cos(z)+ 4*(cos(z))^(3) Cos[3*z]= - 3*Cos[z]+ 4*(Cos[z])^(3) Successful Successful - -
4.21.E32 sin ⁑ ( 4 ⁒ z ) = 8 ⁒ cos 3 ⁑ z ⁒ sin ⁑ z - 4 ⁒ cos ⁑ z ⁒ sin ⁑ z 4 𝑧 8 3 𝑧 𝑧 4 𝑧 𝑧 {\displaystyle{\displaystyle\sin\left(4z\right)=8{\cos^{3}}z\sin z-4\cos z\sin z}} sin(4*z)= 8*(cos(z))^(3)* sin(z)- 4*cos(z)*sin(z) Sin[4*z]= 8*(Cos[z])^(3)* Sin[z]- 4*Cos[z]*Sin[z] Successful Successful - -
4.21.E33 cos ⁑ ( 4 ⁒ z ) = 8 ⁒ cos 4 ⁑ z - 8 ⁒ cos 2 ⁑ z + 1 4 𝑧 8 4 𝑧 8 2 𝑧 1 {\displaystyle{\displaystyle\cos\left(4z\right)=8{\cos^{4}}z-8{\cos^{2}}z+1}} cos(4*z)= 8*(cos(z))^(4)- 8*(cos(z))^(2)+ 1 Cos[4*z]= 8*(Cos[z])^(4)- 8*(Cos[z])^(2)+ 1 Successful Successful - -
4.21.E34 cos ⁑ ( n ⁒ z ) + i ⁒ sin ⁑ ( n ⁒ z ) = ( cos ⁑ z + i ⁒ sin ⁑ z ) n 𝑛 𝑧 𝑖 𝑛 𝑧 superscript 𝑧 𝑖 𝑧 𝑛 {\displaystyle{\displaystyle\cos\left(nz\right)+i\sin\left(nz\right)=(\cos z+i% \sin z)^{n}}} cos(n*z)+ I*sin(n*z)=(cos(z)+ I*sin(z))^(n) Cos[n*z]+ I*Sin[n*z]=(Cos[z]+ I*Sin[z])^(n) Failure Failure Successful Successful
4.21.E35 sin ⁑ ( n ⁒ z ) = 2 n - 1 ⁒ ∏ k = 0 n - 1 sin ⁑ ( z + k ⁒ Ο€ n ) 𝑛 𝑧 superscript 2 𝑛 1 superscript subscript product π‘˜ 0 𝑛 1 𝑧 π‘˜ πœ‹ 𝑛 {\displaystyle{\displaystyle\sin\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\sin% \left(z+\frac{k\pi}{n}\right)}} sin(n*z)= (2)^(n - 1)* product(sin(z +(k*Pi)/(n)), k = 0..n - 1) Sin[n*z]= (2)^(n - 1)* Product[Sin[z +Divide[k*Pi,n]], {k, 0, n - 1}] Failure Successful Skip -
4.21#Ex1 sin ⁑ z = 2 ⁒ t 1 + t 2 𝑧 2 𝑑 1 superscript 𝑑 2 {\displaystyle{\displaystyle\sin z=\frac{2t}{1+t^{2}}}} sin(z)=(2*t)/(1 + (t)^(2)) Sin[z]=Divide[2*t,1 + (t)^(2)] Failure Failure
Fail
1.319645209+.8008956689*I <- {t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.319645209+.1973727279*I <- {t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-2.983425871+.1973727279*I <- {t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-2.983425871+.8008956689*I <- {t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.3196452105315832, 0.8008956683523827] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.3196452105315832, 0.197372728616861] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.9834258721469893, 0.197372728616861] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.9834258721469893, 0.8008956683523827] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21#Ex2 cos ⁑ z = 1 - t 2 1 + t 2 𝑧 1 superscript 𝑑 2 1 superscript 𝑑 2 {\displaystyle{\displaystyle\cos z=\frac{1-t^{2}}{1+t^{2}}}} cos(z)=(1 - (t)^(2))/(1 + (t)^(2)) Cos[z]=Divide[1 - (t)^(2),1 + (t)^(2)] Failure Failure
Fail
1.222026934-1.440804874*I <- {t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.222026934+2.381981344*I <- {t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.222026934-1.440804874*I <- {t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.222026934+2.381981344*I <- {t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.222026932871195, -1.4408048748700928] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.222026932871195, 2.381981345458328] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.222026932871195, -1.4408048748700928] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.222026932871195, 2.381981345458328] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E37 sin ⁑ z = sin ⁑ x ⁒ cosh ⁑ y + i ⁒ cos ⁑ x ⁒ sinh ⁑ y 𝑧 π‘₯ 𝑦 imaginary-unit π‘₯ 𝑦 {\displaystyle{\displaystyle\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y}} sin(z)= sin(x)*cosh(y)+ I*cos(x)*sinh(y) Sin[z]= Sin[x]*Cosh[y]+ I*Cos[x]*Sinh[y] Failure Failure
Fail
.853077958-.3332024445*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-1.014242973-1.657839572*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
-6.320109918-5.110919454*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.748416289+.7908177296*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[0.8530779599233089, -0.33320244491697526] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.014242971876882, -1.6578395715538454] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-6.320109912960862, -5.110919453310433] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.7484162907172458, 0.7908177289090546] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E38 cos ⁑ z = cos ⁑ x ⁒ cosh ⁑ y - i ⁒ sin ⁑ x ⁒ sinh ⁑ y 𝑧 π‘₯ 𝑦 imaginary-unit π‘₯ 𝑦 {\displaystyle{\displaystyle\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y}} cos(z)= cos(x)*cosh(y)- I*sin(x)*sinh(y) Cos[z]= Cos[x]*Cosh[y]- I*Sin[x]*Sinh[y] Failure Failure
Fail
-.4940560329-.9224954029*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-1.693049015+1.140504690*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
-5.099907002+6.518357974*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.9818221171-.842785687*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[-0.49405603343642457, -0.9224954044013453] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.6930490153249411, 1.1405046889875898] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.099906999325039, 6.518357970685734] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.9818221164102445, -0.842785688781432] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E39 tan ⁑ z = sin ⁑ ( 2 ⁒ x ) + i ⁒ sinh ⁑ ( 2 ⁒ y ) cos ⁑ ( 2 ⁒ x ) + cosh ⁑ ( 2 ⁒ y ) 𝑧 2 π‘₯ imaginary-unit 2 𝑦 2 π‘₯ 2 𝑦 {\displaystyle{\displaystyle\tan z=\frac{\sin\left(2x\right)+\mathrm{i}\sinh% \left(2y\right)}{\cos\left(2x\right)+\cosh\left(2y\right)}}} tan(z)=(sin(2*x)+ I*sinh(2*y))/(cos(2*x)+ cosh(2*y)) Tan[z]=Divide[Sin[2*x]+ I*Sinh[2*y],Cos[2*x]+ Cosh[2*y]] Failure Failure
Fail
-.2308812766+.34450810e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
.705848261e-2+.103580521*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.3635417141e-1+.116319149*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.2843295099-.48362120e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[-0.23088127675619197, 0.03445081006213968] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.007058482483423091, 0.1035805212542007] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.03635417128666136, 0.1163191491550224] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.28432950974904503, -0.04836211984008565] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E40 cot ⁑ z = sin ⁑ ( 2 ⁒ x ) - i ⁒ sinh ⁑ ( 2 ⁒ y ) cosh ⁑ ( 2 ⁒ y ) - cos ⁑ ( 2 ⁒ x ) 𝑧 2 π‘₯ imaginary-unit 2 𝑦 2 𝑦 2 π‘₯ {\displaystyle{\displaystyle\cot z=\frac{\sin\left(2x\right)-\mathrm{i}\sinh% \left(2y\right)}{\cosh\left(2y\right)-\cos\left(2x\right)}}} cot(z)=(sin(2*x)- I*sinh(2*y))/(cosh(2*y)- cos(2*x)) Cot[z]=Divide[Sin[2*x]- I*Sinh[2*y],Cosh[2*y]- Cos[2*x]] Failure Failure
Fail
-.1849879852-.249484083e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-.16417892e-3+.913666750e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.2813503899e-1+.1049663965*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.2040171895-.716327538e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[-0.18498798535387256, -0.024948408389711685] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.6417903322245991*^-4, 0.09136667517255426] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.02813503889543659, 0.10496639594574086] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.20401718940971514, -0.07163275379178491] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E41 | sin ⁑ z | = ( sin 2 ⁑ x + sinh 2 ⁑ y ) 1 / 2 𝑧 superscript 2 π‘₯ 2 𝑦 1 2 {\displaystyle{\displaystyle|\sin z|=({\sin^{2}}x+{\sinh^{2}}y)^{1/2}}} abs(sin(z))=((sin(x))^(2)+ (sinh(y))^(2))^(1/ 2) Abs[Sin[z]]=((Sin[x])^(2)+ (Sinh[y])^(2))^(1/ 2) Failure Failure
Fail
.727197533 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-1.550602076 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
-7.880559200 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.686687095 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
0.7271975341555692 <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-1.5506020747613944 <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-7.880559199441702 <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.6866870962353082 <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E41 ( sin 2 ⁑ x + sinh 2 ⁑ y ) 1 / 2 = ( 1 2 ⁒ ( cosh ⁑ ( 2 ⁒ y ) - cos ⁑ ( 2 ⁒ x ) ) ) 1 / 2 superscript 2 π‘₯ 2 𝑦 1 2 superscript 1 2 2 𝑦 2 π‘₯ 1 2 {\displaystyle{\displaystyle({\sin^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2% }\left(\cosh\left(2y\right)-\cos\left(2x\right)\right)\right)^{1/2}}} ((sin(x))^(2)+ (sinh(y))^(2))^(1/ 2)=((1)/(2)*(cosh(2*y)- cos(2*x)))^(1/ 2) ((Sin[x])^(2)+ (Sinh[y])^(2))^(1/ 2)=(Divide[1,2]*(Cosh[2*y]- Cos[2*x]))^(1/ 2) Successful Successful - -
4.21.E42 | cos ⁑ z | = ( cos 2 ⁑ x + sinh 2 ⁑ y ) 1 / 2 𝑧 superscript 2 π‘₯ 2 𝑦 1 2 {\displaystyle{\displaystyle|\cos z|=({\cos^{2}}x+{\sinh^{2}}y)^{1/2}}} abs(cos(z))=((cos(x))^(2)+ (sinh(y))^(2))^(1/ 2) Abs[Cos[z]]=((Cos[x])^(2)+ (Sinh[y])^(2))^(1/ 2) Failure Failure
Fail
.647885813 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-1.725544377 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
-8.091094362 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.694634194 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
0.6478858145183544 <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-1.7255443754392632 <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-8.091094362320007 <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.694634195495673 <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.21.E42 ( cos 2 ⁑ x + sinh 2 ⁑ y ) 1 / 2 = ( 1 2 ⁒ ( cosh ⁑ ( 2 ⁒ y ) + cos ⁑ ( 2 ⁒ x ) ) ) 1 / 2 superscript 2 π‘₯ 2 𝑦 1 2 superscript 1 2 2 𝑦 2 π‘₯ 1 2 {\displaystyle{\displaystyle({\cos^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2% }(\cosh\left(2y\right)+\cos\left(2x\right))\right)^{1/2}}} ((cos(x))^(2)+ (sinh(y))^(2))^(1/ 2)=((1)/(2)*(cosh(2*y)+ cos(2*x)))^(1/ 2) ((Cos[x])^(2)+ (Sinh[y])^(2))^(1/ 2)=(Divide[1,2]*(Cosh[2*y]+ Cos[2*x]))^(1/ 2) Successful Successful - -
4.21.E43 | tan ⁑ z | = ( cosh ⁑ ( 2 ⁒ y ) - cos ⁑ ( 2 ⁒ x ) cosh ⁑ ( 2 ⁒ y ) + cos ⁑ ( 2 ⁒ x ) ) 1 / 2 𝑧 superscript 2 𝑦 2 π‘₯ 2 𝑦 2 π‘₯ 1 2 {\displaystyle{\displaystyle|\tan z|=\left(\frac{\cosh\left(2y\right)-\cos% \left(2x\right)}{\cosh\left(2y\right)+\cos\left(2x\right)}\right)^{1/2}}} abs(tan(z))=((cosh(2*y)- cos(2*x))/(cosh(2*y)+ cos(2*x)))^(1/ 2) Abs[Tan[z]]=(Divide[Cosh[2*y]- Cos[2*x],Cosh[2*y]+ Cos[2*x]])^(1/ 2) Failure Failure
Fail
.1650695e-2 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
.103763936 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.117055546 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-.72745631e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
0.0016506944407532753 <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.10376393520222194 <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.11705554561068499 <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-0.07274563209398122 <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.22.E1 sin ⁑ z = z ⁒ ∏ n = 1 ∞ ( 1 - z 2 n 2 ⁒ Ο€ 2 ) 𝑧 𝑧 superscript subscript product 𝑛 1 1 superscript 𝑧 2 superscript 𝑛 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\sin z=z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n% ^{2}\pi^{2}}\right)}} sin(z)= z*product(1 -((z)^(2))/((n)^(2)* (Pi)^(2)), n = 1..infinity) Sin[z]= z*Product[1 -Divide[(z)^(2),(n)^(2)* (Pi)^(2)], {n, 1, Infinity}] Successful Successful - -
4.22.E2 cos ⁑ z = ∏ n = 1 ∞ ( 1 - 4 ⁒ z 2 ( 2 ⁒ n - 1 ) 2 ⁒ Ο€ 2 ) 𝑧 superscript subscript product 𝑛 1 1 4 superscript 𝑧 2 superscript 2 𝑛 1 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\cos z=\prod_{n=1}^{\infty}\left(1-\frac{4z^{2}}{(% 2n-1)^{2}\pi^{2}}\right)}} cos(z)= product(1 -(4*(z)^(2))/((2*n - 1)^(2)* (Pi)^(2)), n = 1..infinity) Cos[z]= Product[1 -Divide[4*(z)^(2),(2*n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}] Successful Successful - -
4.22.E3 cot ⁑ z = 1 z + 2 ⁒ z ⁒ βˆ‘ n = 1 ∞ 1 z 2 - n 2 ⁒ Ο€ 2 𝑧 1 𝑧 2 𝑧 superscript subscript 𝑛 1 1 superscript 𝑧 2 superscript 𝑛 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\cot z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z% ^{2}-n^{2}\pi^{2}}}} cot(z)=(1)/(z)+ 2*z*sum((1)/((z)^(2)- (n)^(2)* (Pi)^(2)), n = 1..infinity) Cot[z]=Divide[1,z]+ 2*z*Sum[Divide[1,(z)^(2)- (n)^(2)* (Pi)^(2)], {n, 1, Infinity}] Successful Successful - -
4.22.E4 csc 2 ⁑ z = βˆ‘ n = - ∞ ∞ 1 ( z - n ⁒ Ο€ ) 2 2 𝑧 superscript subscript 𝑛 1 superscript 𝑧 𝑛 πœ‹ 2 {\displaystyle{\displaystyle{\csc^{2}}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n% \pi)^{2}}}} (csc(z))^(2)= sum((1)/((z - n*Pi)^(2)), n = - infinity..infinity) (Csc[z])^(2)= Sum[Divide[1,(z - n*Pi)^(2)], {n, - Infinity, Infinity}] Successful Successful - -
4.22.E5 csc ⁑ z = 1 z + 2 ⁒ z ⁒ βˆ‘ n = 1 ∞ ( - 1 ) n z 2 - n 2 ⁒ Ο€ 2 𝑧 1 𝑧 2 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑧 2 superscript 𝑛 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\csc z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)% ^{n}}{z^{2}-n^{2}\pi^{2}}}} csc(z)=(1)/(z)+ 2*z*sum(((- 1)^(n))/((z)^(2)- (n)^(2)* (Pi)^(2)), n = 1..infinity) Csc[z]=Divide[1,z]+ 2*z*Sum[Divide[(- 1)^(n),(z)^(2)- (n)^(2)* (Pi)^(2)], {n, 1, Infinity}] Successful Successful - -
4.23.E1 Arcsin ⁑ z = ∫ 0 z d t ( 1 - t 2 ) 1 / 2 multivalued-inverse-sine 𝑧 superscript subscript 0 𝑧 𝑑 superscript 1 superscript 𝑑 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}z=\int_{0}^{z}\frac{\mathrm{d% }t}{(1-t^{2})^{1/2}}}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, z}]= Integrate[Divide[1,(1 - (t)^(2))^(1/ 2)], {t, 0, z}] Error Failure - Successful
4.23.E2 Arccos ⁑ z = ∫ z 1 d t ( 1 - t 2 ) 1 / 2 multivalued-inverse-cosine 𝑧 superscript subscript 𝑧 1 𝑑 superscript 1 superscript 𝑑 2 1 2 {\displaystyle{\displaystyle\operatorname{Arccos}z=\int_{z}^{1}\frac{\mathrm{d% }t}{(1-t^{2})^{1/2}}}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, z, 1}]= Integrate[Divide[1,(1 - (t)^(2))^(1/ 2)], {t, z, 1}] Error Failure - Skip
4.23.E3 Arctan ⁑ z = ∫ 0 z d t 1 + t 2 multivalued-inverse-tangent 𝑧 superscript subscript 0 𝑧 𝑑 1 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{Arctan}z=\int_{0}^{z}\frac{\mathrm{d% }t}{1+t^{2}}}} Error Integrate[Divide[1, 1+t^2], {t, 0, z}]= Integrate[Divide[1,1 + (t)^(2)], {t, 0, z}] Error Failure - Successful
4.23.E4 Arccsc ⁑ z = Arcsin ⁑ ( 1 / z ) multivalued-inverse-cosecant 𝑧 multivalued-inverse-sine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccsc}z=\operatorname{Arcsin}\left(% 1/z\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, Divide[1,z]}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, 1/ z}] Error Failure - Successful
4.23.E5 Arcsec ⁑ z = Arccos ⁑ ( 1 / z ) multivalued-inverse-secant 𝑧 multivalued-inverse-cosine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arcsec}z=\operatorname{Arccos}\left(% 1/z\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, Divide[1,z], 1}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, 1/ z, 1}] Error Failure - Error
4.23.E6 Arccot ⁑ z = Arctan ⁑ ( 1 / z ) multivalued-inverse-cotangent 𝑧 multivalued-inverse-tangent 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccot}z=\operatorname{Arctan}\left(% 1/z\right)}} Error Integrate[Divide[1, 1+t^2], {t, 0, Divide[1,z]}]= Integrate[Divide[1, 1+t^2], {t, 0, 1/ z}] Error Failure - Successful
4.23.E7 arccsc ⁑ z = arcsin ⁑ ( 1 / z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccsc}z=\operatorname{arcsin}\left(% 1/z\right)}} arccsc(z)= arcsin(1/ z) ArcCsc[z]= ArcSin[1/ z] Failure Successful Successful -
4.23.E8 arcsec ⁑ z = arccos ⁑ ( 1 / z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arcsec}z=\operatorname{arccos}\left(% 1/z\right)}} arcsec(z)= arccos(1/ z) ArcSec[z]= ArcCos[1/ z] Failure Successful Successful -
4.23.E9 arccot ⁑ z = arctan ⁑ ( 1 / z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}z=\operatorname{arctan}\left(% 1/z\right)}} arccot(z)= arctan(1/ z) ArcCot[z]= ArcTan[1/ z] Failure Successful Successful -
4.23.E10 arcsin ⁑ ( - z ) = - arcsin ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\left(-z\right)=-% \operatorname{arcsin}z}} arcsin(- z)= - arcsin(z) ArcSin[- z]= - ArcSin[z] Successful Successful - -
4.23.E11 arccos ⁑ ( - z ) = Ο€ - arccos ⁑ z 𝑧 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\left(-z\right)=\pi-% \operatorname{arccos}z}} arccos(- z)= Pi - arccos(z) ArcCos[- z]= Pi - ArcCos[z] Successful Successful - -
4.23.E12 arctan ⁑ ( - z ) = - arctan ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\left(-z\right)=-% \operatorname{arctan}z}} arctan(- z)= - arctan(z) ArcTan[- z]= - ArcTan[z] Successful Successful - -
4.23.E13 arccsc ⁑ ( - z ) = - arccsc ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arccsc}\left(-z\right)=-% \operatorname{arccsc}z}} arccsc(- z)= - arccsc(z) ArcCsc[- z]= - ArcCsc[z] Successful Successful - -
4.23.E14 arcsec ⁑ ( - z ) = Ο€ - arcsec ⁑ z 𝑧 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arcsec}\left(-z\right)=\pi-% \operatorname{arcsec}z}} arcsec(- z)= Pi - arcsec(z) ArcSec[- z]= Pi - ArcSec[z] Successful Successful - -
4.23.E15 arccot ⁑ ( - z ) = - arccot ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}\left(-z\right)=-% \operatorname{arccot}z}} arccot(- z)= - arccot(z) ArcCot[- z]= - ArcCot[z] Failure Successful - -
4.23.E16 arccos ⁑ z = 1 2 ⁒ Ο€ - arcsin ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}z=\tfrac{1}{2}\pi-% \operatorname{arcsin}z}} arccos(z)=(1)/(2)*Pi - arcsin(z) ArcCos[z]=Divide[1,2]*Pi - ArcSin[z] Failure Successful Successful -
4.23.E17 arcsec ⁑ z = 1 2 ⁒ Ο€ - arccsc ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arcsec}z=\tfrac{1}{2}\pi-% \operatorname{arccsc}z}} arcsec(z)=(1)/(2)*Pi - arccsc(z) ArcSec[z]=Divide[1,2]*Pi - ArcCsc[z] Failure Successful Successful -
4.23.E18 arccot ⁑ z = + 1 2 ⁒ Ο€ - arctan ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}z=+\tfrac{1}{2}\pi-% \operatorname{arctan}z}} arccot(z)= +(1)/(2)*Pi - arctan(z) ArcCot[z]= +Divide[1,2]*Pi - ArcTan[z] Successful Failure - Successful
4.23.E18 arccot ⁑ z = - 1 2 ⁒ Ο€ - arctan ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}z=-\tfrac{1}{2}\pi-% \operatorname{arctan}z}} arccot(z)= -(1)/(2)*Pi - arctan(z) ArcCot[z]= -Divide[1,2]*Pi - ArcTan[z] Failure Failure
Fail
3.141592654 <- {z = 1/2}
Fail
3.141592653589793 <- {Rule[z, Rational[1, 2]]}
4.23.E19 arcsin ⁑ z = - i ⁒ ln ⁑ ( ( 1 - z 2 ) 1 / 2 + i ⁒ z ) 𝑧 𝑖 superscript 1 superscript 𝑧 2 1 2 𝑖 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+% iz\right)}} arcsin(z)= - I*ln((1 - (z)^(2))^(1/ 2)+ I*z) ArcSin[z]= - I*Log[(1 - (z)^(2))^(1/ 2)+ I*z] Failure Successful Error -
4.23.E20 arcsin ⁑ x = 1 2 ⁒ Ο€ + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=\tfrac{1}{2}\pi+i\ln\left((% x^{2}-1)^{1/2}+x\right)}} arcsin(x)=(1)/(2)*Pi + I*ln(((x)^(2)- 1)^(1/ 2)+ x) ArcSin[x]=Divide[1,2]*Pi + I*Log[((x)^(2)- 1)^(1/ 2)+ x] Failure Failure Error Error
4.23.E20 arcsin ⁑ x = 1 2 ⁒ Ο€ - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=\tfrac{1}{2}\pi-i\ln\left((% x^{2}-1)^{1/2}+x\right)}} arcsin(x)=(1)/(2)*Pi - I*ln(((x)^(2)- 1)^(1/ 2)+ x) ArcSin[x]=Divide[1,2]*Pi - I*Log[((x)^(2)- 1)^(1/ 2)+ x] Failure Failure Error Error
4.23.E21 arcsin ⁑ x = - 1 2 ⁒ Ο€ + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=-\tfrac{1}{2}\pi+i\ln\left(% (x^{2}-1)^{1/2}-x\right)}} arcsin(x)= -(1)/(2)*Pi + I*ln(((x)^(2)- 1)^(1/ 2)- x) ArcSin[x]= -Divide[1,2]*Pi + I*Log[((x)^(2)- 1)^(1/ 2)- x] Failure Failure Error Error
4.23.E21 arcsin ⁑ x = - 1 2 ⁒ Ο€ - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=-\tfrac{1}{2}\pi-i\ln\left(% (x^{2}-1)^{1/2}-x\right)}} arcsin(x)= -(1)/(2)*Pi - I*ln(((x)^(2)- 1)^(1/ 2)- x) ArcSin[x]= -Divide[1,2]*Pi - I*Log[((x)^(2)- 1)^(1/ 2)- x] Failure Failure Error Error
4.23.E22 arccos ⁑ z = 1 2 ⁒ Ο€ + i ⁒ ln ⁑ ( ( 1 - z 2 ) 1 / 2 + i ⁒ z ) 𝑧 1 2 πœ‹ 𝑖 superscript 1 superscript 𝑧 2 1 2 𝑖 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((% 1-z^{2})^{1/2}+iz\right)}} arccos(z)=(1)/(2)*Pi + I*ln((1 - (z)^(2))^(1/ 2)+ I*z) ArcCos[z]=Divide[1,2]*Pi + I*Log[(1 - (z)^(2))^(1/ 2)+ I*z] Failure Successful Error -
4.23.E23 arccos ⁑ z = - 2 ⁒ i ⁒ ln ⁑ ( ( 1 + z 2 ) 1 / 2 + i ⁒ ( 1 - z 2 ) 1 / 2 ) 𝑧 2 𝑖 superscript 1 𝑧 2 1 2 𝑖 superscript 1 𝑧 2 1 2 {\displaystyle{\displaystyle\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z% }{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}\right)}} arccos(z)= - 2*I*ln(((1 + z)/(2))^(1/ 2)+ I*((1 - z)/(2))^(1/ 2)) ArcCos[z]= - 2*I*Log[(Divide[1 + z,2])^(1/ 2)+ I*(Divide[1 - z,2])^(1/ 2)] Failure Failure Error Error
4.23.E24 arccos ⁑ x = - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=-i\ln\left((x^{2}-1)^{1/2}+% x\right)}} arccos(x)= - I*ln(((x)^(2)- 1)^(1/ 2)+ x) ArcCos[x]= - I*Log[((x)^(2)- 1)^(1/ 2)+ x] Failure Failure Error Error
4.23.E24 arccos ⁑ x = + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=+i\ln\left((x^{2}-1)^{1/2}+% x\right)}} arccos(x)= + I*ln(((x)^(2)- 1)^(1/ 2)+ x) ArcCos[x]= + I*Log[((x)^(2)- 1)^(1/ 2)+ x] Failure Failure Error Error
4.23.E25 arccos ⁑ x = Ο€ - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=\pi-i\ln\left((x^{2}-1)^{1/% 2}-x\right)}} arccos(x)= Pi - I*ln(((x)^(2)- 1)^(1/ 2)- x) ArcCos[x]= Pi - I*Log[((x)^(2)- 1)^(1/ 2)- x] Failure Failure Error Error
4.23.E25 arccos ⁑ x = Ο€ + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=\pi+i\ln\left((x^{2}-1)^{1/% 2}-x\right)}} arccos(x)= Pi + I*ln(((x)^(2)- 1)^(1/ 2)- x) ArcCos[x]= Pi + I*Log[((x)^(2)- 1)^(1/ 2)- x] Failure Failure Error Error
4.23.E26 arctan ⁑ z = i 2 ⁒ ln ⁑ ( i + z i - z ) 𝑧 𝑖 2 𝑖 𝑧 𝑖 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i% +z}{i-z}\right)}} arctan(z)=(I)/(2)*ln((I + z)/(I - z)) ArcTan[z]=Divide[I,2]*Log[Divide[I + z,I - z]] Failure Failure Error Error
4.23.E27 arctan ⁑ ( i ⁒ y ) = + 1 2 ⁒ Ο€ + i 2 ⁒ ln ⁑ ( y + 1 y - 1 ) 𝑖 𝑦 1 2 πœ‹ 𝑖 2 𝑦 1 𝑦 1 {\displaystyle{\displaystyle\operatorname{arctan}\left(iy\right)=+\frac{1}{2}% \pi+\frac{i}{2}\ln\left(\frac{y+1}{y-1}\right)}} arctan(I*y)= +(1)/(2)*Pi +(I)/(2)*ln((y + 1)/(y - 1)) ArcTan[I*y]= +Divide[1,2]*Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]] Failure Failure Error Error
4.23.E27 arctan ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ + i 2 ⁒ ln ⁑ ( y + 1 y - 1 ) 𝑖 𝑦 1 2 πœ‹ 𝑖 2 𝑦 1 𝑦 1 {\displaystyle{\displaystyle\operatorname{arctan}\left(iy\right)=-\frac{1}{2}% \pi+\frac{i}{2}\ln\left(\frac{y+1}{y-1}\right)}} arctan(I*y)= -(1)/(2)*Pi +(I)/(2)*ln((y + 1)/(y - 1)) ArcTan[I*y]= -Divide[1,2]*Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]] Failure Failure Error Error
4.23.E28 z = sin ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\sin w}} z = sin(w) z = Sin[w] Failure Failure
Fail
-.737321978+1.112452092*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-.737321978-1.715975032*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-3.565749102-1.715975032*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-3.565749102+1.112452092*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.7373219789661911, 1.1124520925053343] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.7373219789661911, -1.715975032240856] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.565749103712381, -1.715975032240856] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.565749103712381, 1.1124520925053343] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E29 z = cos ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\cos w}} z = cos(w) z = Cos[w] Failure Failure
Fail
1.074539570+3.325606671*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.074539570+.497179547*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.753887554+.497179547*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.753887554+3.325606671*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.0745395706783705, 3.3256066725373055] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.0745395706783705, 0.4971795477911152] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7538875540678198, 0.4971795477911152] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7538875540678198, 3.3256066725373055] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E30 z = tan ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\tan w}} z = tan(w) z = Tan[w] Failure Failure
Fail
1.373342253+.295839425*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.373342253-2.532587699*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.455084871-2.532587699*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.455084871+.295839425*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.3733422538097753, 0.2958394249722609] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.3733422538097753, -2.5325876997739294] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.455084870936415, -2.5325876997739294] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.455084870936415, 0.2958394249722609] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E31 w = Arcsin ⁑ z 𝑀 multivalued-inverse-sine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arcsin}z}} Error w = Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, z}] Error Failure -
Fail
Complex[0.6905247538249891, 0.022188930362652126] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.6905247538249891, 2.806238194383538] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.1379023709212013, 2.806238194383538] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.1379023709212013, 0.022188930362652126] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E31 Arcsin ⁑ z = ( - 1 ) k ⁒ arcsin ⁑ z + k ⁒ Ο€ multivalued-inverse-sine 𝑧 superscript 1 π‘˜ 𝑧 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arcsin}z=(-1)^{k}\operatorname{% arcsin}z+k\pi}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, z}]=(- 1)^(k)* ArcSin[z]+ k*Pi Error Failure -
Fail
Complex[-1.694215036493581, 2.784049264020886] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-6.283185307179586 <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-7.9774003436731675, 2.784049264020886] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.694215036493581, -2.784049264020886] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E32 w = Arccos ⁑ z 𝑀 multivalued-inverse-cosine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arccos}z}} Error w = Integrate[Divide[1, (1-t^2)^(1/2)], {t, z, 1}] Error Failure - Successful
4.23.E32 Arccos ⁑ z = + arccos ⁑ z + 2 ⁒ k ⁒ Ο€ multivalued-inverse-cosine 𝑧 𝑧 2 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arccos}z=+\operatorname{arccos}z+2k% \pi}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, z, 1}]= + ArcCos[z]+ 2*k*Pi Error Failure - Successful
4.23.E32 Arccos ⁑ z = - arccos ⁑ z + 2 ⁒ k ⁒ Ο€ multivalued-inverse-cosine 𝑧 𝑧 2 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arccos}z=-\operatorname{arccos}z+2k% \pi}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, z, 1}]= - ArcCos[z]+ 2*k*Pi Error Failure - Successful
4.23.E33 w = Arctan ⁑ z 𝑀 multivalued-inverse-tangent 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arctan}z}} Error w = Integrate[Divide[1, 1+t^2], {t, 0, z}] Error Failure -
Fail
Complex[0.950565953372289, 1.4142135623730951] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
Complex[0.950565953372289, -1.4142135623730951] <- {Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
Complex[-1.8778611713739013, -1.4142135623730951] <- {Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
Complex[-1.8778611713739013, 1.4142135623730951] <- {Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
4.23.E33 Arctan ⁑ z = arctan ⁑ z + k ⁒ Ο€ multivalued-inverse-tangent 𝑧 𝑧 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arctan}z=\operatorname{arctan}z+k\pi}} Error Integrate[Divide[1, 1+t^2], {t, 0, z}]= ArcTan[z]+ k*Pi Error Failure -
Fail
-3.141592653589793 <- {Rule[k, 1], Rule[z, Rational[1, 2]]}
-6.283185307179586 <- {Rule[k, 2], Rule[z, Rational[1, 2]]}
-9.42477796076938 <- {Rule[k, 3], Rule[z, Rational[1, 2]]}
4.23.E34 arcsin ⁑ z = arcsin ⁑ Ξ² + i ⁒ sign ⁑ ( y ) ⁒ ln ⁑ ( Ξ± + ( Ξ± 2 - 1 ) 1 / 2 ) 𝑧 𝛽 imaginary-unit sign 𝑦 𝛼 superscript superscript 𝛼 2 1 1 2 {\displaystyle{\displaystyle\operatorname{arcsin}z=\operatorname{arcsin}\beta+% \mathrm{i}\operatorname{sign}\left(y\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2% }\right)}} arcsin(z)= arcsin(beta)+ I*signum(y)*ln(alpha +((alpha)^(2)- 1)^(1/ 2)) ArcSin[z]= ArcSin[\[Beta]]+ I*Sign[y]*Log[\[Alpha]+((\[Alpha])^(2)- 1)^(1/ 2)] Failure Failure
Fail
.8471075183-1.392024632*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 1}
.8471075183-1.392024632*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 2}
.8471075183-1.392024632*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 3}
.8471075183-4.176073896*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
Fail
Complex[0.8471075182467906, -1.3920246320104428] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8471075182467906, -1.3920246320104428] <- {Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8471075182467906, -1.3920246320104428] <- {Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8471075182467906, 1.3920246320104432] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E35 arccos ⁑ z = arccos ⁑ Ξ² - i ⁒ sign ⁑ ( y ) ⁒ ln ⁑ ( Ξ± + ( Ξ± 2 - 1 ) 1 / 2 ) 𝑧 𝛽 imaginary-unit sign 𝑦 𝛼 superscript superscript 𝛼 2 1 1 2 {\displaystyle{\displaystyle\operatorname{arccos}z=\operatorname{arccos}\beta-% \mathrm{i}\operatorname{sign}\left(y\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2% }\right)}} arccos(z)= arccos(beta)- I*signum(y)*ln(alpha +((alpha)^(2)- 1)^(1/ 2)) ArcCos[z]= ArcCos[\[Beta]]- I*Sign[y]*Log[\[Alpha]+((\[Alpha])^(2)- 1)^(1/ 2)] Failure Failure
Fail
-.8471075183+1.392024632*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 1}
-.8471075183+1.392024632*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 2}
-.8471075183+1.392024632*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 3}
-.8471075183+4.176073896*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
Fail
Complex[-0.8471075182467906, 1.3920246320104428] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.8471075182467906, 1.3920246320104428] <- {Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.8471075182467906, 1.3920246320104428] <- {Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.8471075182467906, -1.3920246320104432] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ±, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ², Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E36 arctan ⁑ z = 1 2 ⁒ arctan ⁑ ( 2 ⁒ x 1 - x 2 - y 2 ) + 1 4 ⁒ i ⁒ ln ⁑ ( x 2 + ( y + 1 ) 2 x 2 + ( y - 1 ) 2 ) 𝑧 1 2 2 π‘₯ 1 superscript π‘₯ 2 superscript 𝑦 2 1 4 𝑖 superscript π‘₯ 2 superscript 𝑦 1 2 superscript π‘₯ 2 superscript 𝑦 1 2 {\displaystyle{\displaystyle\operatorname{arctan}z=\tfrac{1}{2}\operatorname{% arctan}\left(\frac{2x}{1-x^{2}-y^{2}}\right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}% +(y+1)^{2}}{x^{2}+(y-1)^{2}}\right)}} arctan(z)=(1)/(2)*arctan((2*x)/(1 - (x)^(2)- (y)^(2)))+(1)/(4)*I*ln(((x)^(2)+(y + 1)^(2))/((x)^(2)+(y - 1)^(2))) ArcTan[z]=Divide[1,2]*ArcTan[Divide[2*x,1 - (x)^(2)- (y)^(2)]]+Divide[1,4]*I*Log[Divide[(x)^(2)+(y + 1)^(2),(x)^(2)+(y - 1)^(2)]] Failure Failure
Fail
1.746385980-.817820081e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
1.424635426-.817820081e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
1.302146094+.146336119e-1*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
1.585510703+.1472906747*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[1.746385980479988, -0.081782008243384] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4246354260833458, -0.081782008243384] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.3021460945199137, 0.014633611959612158] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.5855107032816669, 0.14729067472515475] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.23.E39 gd ⁑ ( x ) = ∫ 0 x sech ⁑ t ⁒ d t Gudermannian π‘₯ superscript subscript 0 π‘₯ 𝑑 𝑑 {\displaystyle{\displaystyle\operatorname{gd}\left(x\right)=\int_{0}^{x}% \operatorname{sech}t\mathrm{d}t}} arctan(sinh(x))= int(sech(t), t = 0..x) Gudermannian[x]= Integrate[Sech[t], {t, 0, x}] Successful Failure - Successful
4.23.E40 2 ⁒ arctan ⁑ ( e x ) - 1 2 ⁒ Ο€ = arcsin ⁑ ( tanh ⁑ x ) 2 superscript 𝑒 π‘₯ 1 2 πœ‹ π‘₯ {\displaystyle{\displaystyle 2\operatorname{arctan}\left(e^{x}\right)-\tfrac{1% }{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)}} 2*arctan(exp(x))-(1)/(2)*Pi = arcsin(tanh(x)) 2*ArcTan[Exp[x]]-Divide[1,2]*Pi = ArcSin[Tanh[x]] Error Error - -
4.23.E40 arccsc ⁑ ( coth ⁑ x ) = arccos ⁑ ( sech ⁑ x ) hyperbolic-cotangent π‘₯ π‘₯ {\displaystyle{\displaystyle\operatorname{arccsc}\left(\coth x\right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)}} arccsc(coth(x))= arccos(sech(x)) ArcCsc[Coth[x]]= ArcCos[Sech[x]] Error Error - -
4.23.E40 arcsec ⁑ ( cosh ⁑ x ) = arctan ⁑ ( sinh ⁑ x ) π‘₯ π‘₯ {\displaystyle{\displaystyle\operatorname{arcsec}\left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)}} arcsec(cosh(x))= arctan(sinh(x)) ArcSec[Cosh[x]]= ArcTan[Sinh[x]] Error Error - -
4.23.E40 arctan ⁑ ( sinh ⁑ x ) = arccot ⁑ ( csch ⁑ x ) π‘₯ π‘₯ {\displaystyle{\displaystyle\operatorname{arctan}\left(\sinh x\right)=% \operatorname{arccot}\left(\operatorname{csch}x\right)}} arctan(sinh(x))= arccot(csch(x)) ArcTan[Sinh[x]]= ArcCot[Csch[x]] Failure Successful Skip -
4.23.E41 gd - 1 ⁑ ( x ) = ∫ 0 x sec ⁑ t ⁒ d t inverse-Gudermannian π‘₯ superscript subscript 0 π‘₯ 𝑑 𝑑 {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\int_{0}^{x% }\sec t\mathrm{d}t}} arctanh(sin(x))= int(sec(t), t = 0..x) InverseGudermannian[x]= Integrate[Sec[t], {t, 0, x}] Successful Failure - Successful
4.23.E42 gd - 1 ⁑ ( x ) = ln ⁑ tan ⁑ ( 1 2 ⁒ x + 1 4 ⁒ Ο€ ) inverse-Gudermannian π‘₯ 1 2 π‘₯ 1 4 πœ‹ {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\ln\tan% \left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)}} arctanh(sin(x))= ln(tan((1)/(2)*x +(1)/(4)*Pi)) InverseGudermannian[x]= Log[Tan[Divide[1,2]*x +Divide[1,4]*Pi]] Failure Successful
Fail
.2e-8-3.141592654*I <- {x = 2}
.9e-9-3.141592654*I <- {x = 3}
 -
4.23.E42 ln ⁑ tan ⁑ ( 1 2 ⁒ x + 1 4 ⁒ Ο€ ) = ln ⁑ ( sec ⁑ x + tan ⁑ x ) 1 2 π‘₯ 1 4 πœ‹ π‘₯ π‘₯ {\displaystyle{\displaystyle\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)=% \ln\left(\sec x+\tan x\right)}} ln(tan((1)/(2)*x +(1)/(4)*Pi))= ln(sec(x)+ tan(x)) Log[Tan[Divide[1,2]*x +Divide[1,4]*Pi]]= Log[Sec[x]+ Tan[x]] Successful Successful - -
4.23.E42 ln ⁑ ( sec ⁑ x + tan ⁑ x ) = arcsinh ⁑ ( tan ⁑ x ) π‘₯ π‘₯ hyperbolic-inverse-sine π‘₯ {\displaystyle{\displaystyle\ln\left(\sec x+\tan x\right)=\operatorname{% arcsinh}\left(\tan x\right)}} ln(sec(x)+ tan(x))= arcsinh(tan(x)) Log[Sec[x]+ Tan[x]]= ArcSinh[Tan[x]] Failure Failure Skip
Fail
Complex[3.046904887125347, 3.141592653589793] <- {Rule[x, 2]}
Complex[0.28413631677879303, 3.141592653589793] <- {Rule[x, 3]}
4.23.E42 arcsinh ⁑ ( tan ⁑ x ) = arccsch ⁑ ( cot ⁑ x ) hyperbolic-inverse-sine π‘₯ hyperbolic-inverse-cosecant π‘₯ {\displaystyle{\displaystyle\operatorname{arcsinh}\left(\tan x\right)=% \operatorname{arccsch}\left(\cot x\right)}} arcsinh(tan(x))= arccsch(cot(x)) ArcSinh[Tan[x]]= ArcCsch[Cot[x]] Failure Successful Skip -
4.23.E42 arccsch ⁑ ( cot ⁑ x ) = arccosh ⁑ ( sec ⁑ x ) hyperbolic-inverse-cosecant π‘₯ hyperbolic-inverse-cosine π‘₯ {\displaystyle{\displaystyle\operatorname{arccsch}\left(\cot x\right)=% \operatorname{arccosh}\left(\sec x\right)}} arccsch(cot(x))= arccosh(sec(x)) ArcCsch[Cot[x]]= ArcCosh[Sec[x]] Failure Failure Skip
Fail
Complex[-3.046904887125347, -3.141592653589793] <- {Rule[x, 2]}
Complex[-0.2841363167787935, -3.141592653589793] <- {Rule[x, 3]}
4.23.E42 arccosh ⁑ ( sec ⁑ x ) = arcsech ⁑ ( cos ⁑ x ) hyperbolic-inverse-cosine π‘₯ hyperbolic-inverse-secant π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}\left(\sec x\right)=% \operatorname{arcsech}\left(\cos x\right)}} arccosh(sec(x))= arcsech(cos(x)) ArcCosh[Sec[x]]= ArcSech[Cos[x]] Failure Successful Skip -
4.23.E42 arcsech ⁑ ( cos ⁑ x ) = arctanh ⁑ ( sin ⁑ x ) hyperbolic-inverse-secant π‘₯ hyperbolic-inverse-tangent π‘₯ {\displaystyle{\displaystyle\operatorname{arcsech}\left(\cos x\right)=% \operatorname{arctanh}\left(\sin x\right)}} arcsech(cos(x))= arctanh(sin(x)) ArcSech[Cos[x]]= ArcTanh[Sin[x]] Failure Failure Skip
Fail
Complex[0.0, 3.141592653589793] <- {Rule[x, 2]}
Complex[5.273559366969494*^-16, 3.141592653589793] <- {Rule[x, 3]}
4.23.E42 arctanh ⁑ ( sin ⁑ x ) = arccoth ⁑ ( csc ⁑ x ) hyperbolic-inverse-tangent π‘₯ hyperbolic-inverse-cotangent π‘₯ {\displaystyle{\displaystyle\operatorname{arctanh}\left(\sin x\right)=% \operatorname{arccoth}\left(\csc x\right)}} arctanh(sin(x))= arccoth(csc(x)) ArcTanh[Sin[x]]= ArcCoth[Csc[x]] Failure Successful Skip -
4.24.E7 d d z ⁑ arcsin ⁑ z = ( 1 - z 2 ) - 1 / 2 derivative 𝑧 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsin% }z=(1-z^{2})^{-1/2}}} diff(arcsin(z), z)=(1 - (z)^(2))^(- 1/ 2) D[ArcSin[z], z]=(1 - (z)^(2))^(- 1/ 2) Successful Successful - -
4.24.E8 d d z ⁑ arccos ⁑ z = - ( 1 - z 2 ) - 1 / 2 derivative 𝑧 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccos% }z=-(1-z^{2})^{-1/2}}} diff(arccos(z), z)= -(1 - (z)^(2))^(- 1/ 2) D[ArcCos[z], z]= -(1 - (z)^(2))^(- 1/ 2) Successful Successful - -
4.24.E9 d d z ⁑ arctan ⁑ z = 1 1 + z 2 derivative 𝑧 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctan% }z=\frac{1}{1+z^{2}}}} diff(arctan(z), z)=(1)/(1 + (z)^(2)) D[ArcTan[z], z]=Divide[1,1 + (z)^(2)] Successful Successful - -
4.24.E10 d d z ⁑ arccsc ⁑ z = - 1 z ⁒ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc% }z=-\frac{1}{z(z^{2}-1)^{1/2}}}} diff(arccsc(z), z)= -(1)/(z*((z)^(2)- 1)^(1/ 2)) D[ArcCsc[z], z]= -Divide[1,z*((z)^(2)- 1)^(1/ 2)] Failure Failure Successful Successful
4.24.E10 d d z ⁑ arccsc ⁑ z = + 1 z ⁒ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc% }z=+\frac{1}{z(z^{2}-1)^{1/2}}}} diff(arccsc(z), z)= +(1)/(z*((z)^(2)- 1)^(1/ 2)) D[ArcCsc[z], z]= +Divide[1,z*((z)^(2)- 1)^(1/ 2)] Failure Failure
Fail
4.618802153*I <- {z = 1/2}
Fail
Complex[0.0, 4.618802153517007] <- {Rule[z, Rational[1, 2]]}
4.24.E11 d d z ⁑ arcsec ⁑ z = + 1 z ⁒ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec% }z=+\frac{1}{z(z^{2}-1)^{1/2}}}} diff(arcsec(z), z)= +(1)/(z*((z)^(2)- 1)^(1/ 2)) D[ArcSec[z], z]= +Divide[1,z*((z)^(2)- 1)^(1/ 2)] Failure Failure Successful Successful
4.24.E11 d d z ⁑ arcsec ⁑ z = - 1 z ⁒ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec% }z=-\frac{1}{z(z^{2}-1)^{1/2}}}} diff(arcsec(z), z)= -(1)/(z*((z)^(2)- 1)^(1/ 2)) D[ArcSec[z], z]= -Divide[1,z*((z)^(2)- 1)^(1/ 2)] Failure Failure
Fail
-4.618802153*I <- {z = 1/2}
Fail
Complex[0.0, -4.618802153517007] <- {Rule[z, Rational[1, 2]]}
4.24.E12 d d z ⁑ arccot ⁑ z = - 1 1 + z 2 derivative 𝑧 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccot% }z=-\frac{1}{1+z^{2}}}} diff(arccot(z), z)= -(1)/(1 + (z)^(2)) D[ArcCot[z], z]= -Divide[1,1 + (z)^(2)] Successful Successful - -
4.24.E13 Arcsin ⁑ u + Arcsin ⁑ v = Arcsin ⁑ ( u ⁒ ( 1 - v 2 ) 1 / 2 + v ⁒ ( 1 - u 2 ) 1 / 2 ) multivalued-inverse-sine 𝑒 multivalued-inverse-sine 𝑣 multivalued-inverse-sine 𝑒 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑒 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arcsin}v=% \operatorname{Arcsin}\left(u(1-v^{2})^{1/2}+v(1-u^{2})^{1/2}\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u}]+ Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, v}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u*(1 - (v)^(2))^(1/ 2)+ v*(1 - (u)^(2))^(1/ 2)}] Error Failure - Successful
4.24.E13 Arcsin ⁑ u - Arcsin ⁑ v = Arcsin ⁑ ( u ⁒ ( 1 - v 2 ) 1 / 2 - v ⁒ ( 1 - u 2 ) 1 / 2 ) multivalued-inverse-sine 𝑒 multivalued-inverse-sine 𝑣 multivalued-inverse-sine 𝑒 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑒 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arcsin}v=% \operatorname{Arcsin}\left(u(1-v^{2})^{1/2}-v(1-u^{2})^{1/2}\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u}]- Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, v}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u*(1 - (v)^(2))^(1/ 2)- v*(1 - (u)^(2))^(1/ 2)}] Error Failure - Successful
4.24.E14 Arccos ⁑ u + Arccos ⁑ v = Arccos ⁑ ( u ⁒ v - ( ( 1 - u 2 ) ⁒ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-cosine 𝑒 multivalued-inverse-cosine 𝑣 multivalued-inverse-cosine 𝑒 𝑣 superscript 1 superscript 𝑒 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arccos}u+\operatorname{Arccos}v=% \operatorname{Arccos}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, u, 1}]+ Integrate[Divide[1, (1-t^2)^(1/2)], {t, v, 1}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/ 2), 1}] Error Failure - Error
4.24.E14 Arccos ⁑ u - Arccos ⁑ v = Arccos ⁑ ( u ⁒ v + ( ( 1 - u 2 ) ⁒ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-cosine 𝑒 multivalued-inverse-cosine 𝑣 multivalued-inverse-cosine 𝑒 𝑣 superscript 1 superscript 𝑒 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arccos}u-\operatorname{Arccos}v=% \operatorname{Arccos}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, u, 1}]- Integrate[Divide[1, (1-t^2)^(1/2)], {t, v, 1}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/ 2), 1}] Error Failure - Error
4.24.E15 Arctan ⁑ u + Arctan ⁑ v = Arctan ⁑ ( u + v 1 - u ⁒ v ) multivalued-inverse-tangent 𝑒 multivalued-inverse-tangent 𝑣 multivalued-inverse-tangent 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arctan}v=% \operatorname{Arctan}\left(\frac{u+v}{1-uv}\right)}} Error Integrate[Divide[1, 1+t^2], {t, 0, u}]+ Integrate[Divide[1, 1+t^2], {t, 0, v}]= Integrate[Divide[1, 1+t^2], {t, 0, Divide[u + v,1 - u*v]}] Error Failure -
Fail
Complex[3.141592653589793, 3.3306690738754696*^-16] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.141592653589793, 0.0] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.141592653589793, 0.0] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.141592653589793, -3.3306690738754696*^-16] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.24.E15 Arctan ⁑ u - Arctan ⁑ v = Arctan ⁑ ( u - v 1 + u ⁒ v ) multivalued-inverse-tangent 𝑒 multivalued-inverse-tangent 𝑣 multivalued-inverse-tangent 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arctan}v=% \operatorname{Arctan}\left(\frac{u-v}{1+uv}\right)}} Error Integrate[Divide[1, 1+t^2], {t, 0, u}]- Integrate[Divide[1, 1+t^2], {t, 0, v}]= Integrate[Divide[1, 1+t^2], {t, 0, Divide[u - v,1 + u*v]}] Error Failure -
Fail
Complex[3.141592653589793, 3.3306690738754696*^-16] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.141592653589793, 0.0] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.141592653589793, 0.0] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.141592653589793, -3.3306690738754696*^-16] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.24.E16 Arcsin ⁑ u + Arccos ⁑ v = Arcsin ⁑ ( u ⁒ v + ( ( 1 - u 2 ) ⁒ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-sine 𝑒 multivalued-inverse-cosine 𝑣 multivalued-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arccos}v=% \operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u}]+ Integrate[Divide[1, (1-t^2)^(1/2)], {t, v, 1}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/ 2)}] Error Failure - Skip
4.24.E16 Arcsin ⁑ u - Arccos ⁑ v = Arcsin ⁑ ( u ⁒ v - ( ( 1 - u 2 ) ⁒ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-sine 𝑒 multivalued-inverse-cosine 𝑣 multivalued-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arccos}v=% \operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u}]- Integrate[Divide[1, (1-t^2)^(1/2)], {t, v, 1}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/ 2)}] Error Failure - Skip
4.24.E16 Arcsin ⁑ ( u ⁒ v + ( ( 1 - u 2 ) ⁒ ( 1 - v 2 ) ) 1 / 2 ) = Arccos ⁑ ( v ⁒ ( 1 - u 2 ) 1 / 2 - u ⁒ ( 1 - v 2 ) 1 / 2 ) multivalued-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 1 superscript 𝑣 2 1 2 multivalued-inverse-cosine 𝑣 superscript 1 superscript 𝑒 2 1 2 𝑒 superscript 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))% ^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}-u(1-v^{2})^{1/2}% \right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/ 2)}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, v*(1 - (u)^(2))^(1/ 2)- u*(1 - (v)^(2))^(1/ 2), 1}] Error Failure - Successful
4.24.E16 Arcsin ⁑ ( u ⁒ v - ( ( 1 - u 2 ) ⁒ ( 1 - v 2 ) ) 1 / 2 ) = Arccos ⁑ ( v ⁒ ( 1 - u 2 ) 1 / 2 + u ⁒ ( 1 - v 2 ) 1 / 2 ) multivalued-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 1 superscript 𝑣 2 1 2 multivalued-inverse-cosine 𝑣 superscript 1 superscript 𝑒 2 1 2 𝑒 superscript 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))% ^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}+u(1-v^{2})^{1/2}% \right)}} Error Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/ 2)}]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, v*(1 - (u)^(2))^(1/ 2)+ u*(1 - (v)^(2))^(1/ 2), 1}] Error Failure - Error
4.24.E17 Arctan ⁑ u + Arccot ⁑ v = Arctan ⁑ ( u ⁒ v + 1 v - u ) multivalued-inverse-tangent 𝑒 multivalued-inverse-cotangent 𝑣 multivalued-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 {\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arccot}v=% \operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)}} Error Integrate[Divide[1, 1+t^2], {t, 0, u}]+ Integrate[Divide[1, 1+t^2], {t, 0, Divide[1,v]}]= Integrate[Divide[1, 1+t^2], {t, 0, Divide[u*v + 1,v - u]}] Error Failure - Successful
4.24.E17 Arctan ⁑ u - Arccot ⁑ v = Arctan ⁑ ( u ⁒ v - 1 v + u ) multivalued-inverse-tangent 𝑒 multivalued-inverse-cotangent 𝑣 multivalued-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 {\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arccot}v=% \operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)}} Error Integrate[Divide[1, 1+t^2], {t, 0, u}]- Integrate[Divide[1, 1+t^2], {t, 0, Divide[1,v]}]= Integrate[Divide[1, 1+t^2], {t, 0, Divide[u*v - 1,v + u]}] Error Failure - Error
4.24.E17 Arctan ⁑ ( u ⁒ v + 1 v - u ) = Arccot ⁑ ( v - u u ⁒ v + 1 ) multivalued-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 multivalued-inverse-cotangent 𝑣 𝑒 𝑒 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)% =\operatorname{Arccot}\left(\frac{v-u}{uv+1}\right)}} Error Integrate[Divide[1, 1+t^2], {t, 0, Divide[u*v + 1,v - u]}]= Integrate[Divide[1, 1+t^2], {t, 0, Divide[1,Divide[v - u,u*v + 1]]}] Error Failure - Successful
4.24.E17 Arctan ⁑ ( u ⁒ v - 1 v + u ) = Arccot ⁑ ( v + u u ⁒ v - 1 ) multivalued-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 multivalued-inverse-cotangent 𝑣 𝑒 𝑒 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)% =\operatorname{Arccot}\left(\frac{v+u}{uv-1}\right)}} Error Integrate[Divide[1, 1+t^2], {t, 0, Divide[u*v - 1,v + u]}]= Integrate[Divide[1, 1+t^2], {t, 0, Divide[1,Divide[v + u,u*v - 1]]}] Error Failure - Error
4.26.E1 ∫ sin ⁑ x ⁒ d x = - cos ⁑ x π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\sin x\mathrm{d}x=-\cos x}} int(sin(x), x)= - cos(x) Integrate[Sin[x], x]= - Cos[x] Successful Successful - -
4.26.E2 ∫ cos ⁑ x ⁒ d x = sin ⁑ x π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\cos x\mathrm{d}x=\sin x}} int(cos(x), x)= sin(x) Integrate[Cos[x], x]= Sin[x] Successful Successful - -
4.26.E3 ∫ tan ⁑ x ⁒ d x = - ln ⁑ ( cos ⁑ x ) π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\tan x\mathrm{d}x=-\ln\left(\cos x\right)}} int(tan(x), x)= - ln(cos(x)) Integrate[Tan[x], x]= - Log[Cos[x]] Successful Successful - -
4.26.E4 ∫ csc ⁑ x ⁒ d x = ln ⁑ ( tan ⁑ 1 2 ⁒ x ) π‘₯ π‘₯ 1 2 π‘₯ {\displaystyle{\displaystyle\int\csc x\mathrm{d}x=\ln\left(\tan\tfrac{1}{2}x% \right)}} int(csc(x), x)= ln(tan((1)/(2)*x)) Integrate[Csc[x], x]= Log[Tan[Divide[1,2]*x]] Failure Successful Skip -
4.26.E5 ∫ sec ⁑ x ⁒ d x = gd - 1 ⁑ ( x ) π‘₯ π‘₯ inverse-Gudermannian π‘₯ {\displaystyle{\displaystyle\int\sec x\mathrm{d}x={\operatorname{gd}^{-1}}% \left(x\right)}} int(sec(x), x)= arctanh(sin(x)) Integrate[Sec[x], x]= InverseGudermannian[x] Failure Failure Skip Successful
4.26.E6 ∫ cot ⁑ x ⁒ d x = ln ⁑ ( sin ⁑ x ) π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\cot x\mathrm{d}x=\ln\left(\sin x\right)}} int(cot(x), x)= ln(sin(x)) Integrate[Cot[x], x]= Log[Sin[x]] Successful Successful - -
4.26.E7 ∫ e a ⁒ x ⁒ sin ⁑ ( b ⁒ x ) ⁒ d x = e a ⁒ x a 2 + b 2 ⁒ ( a ⁒ sin ⁑ ( b ⁒ x ) - b ⁒ cos ⁑ ( b ⁒ x ) ) superscript 𝑒 π‘Ž π‘₯ 𝑏 π‘₯ π‘₯ superscript 𝑒 π‘Ž π‘₯ superscript π‘Ž 2 superscript 𝑏 2 π‘Ž 𝑏 π‘₯ 𝑏 𝑏 π‘₯ {\displaystyle{\displaystyle\int e^{ax}\sin\left(bx\right)\mathrm{d}x=\frac{e^% {ax}}{a^{2}+b^{2}}(a\sin\left(bx\right)-b\cos\left(bx\right))}} int(exp(a*x)*sin(b*x), x)=(exp(a*x))/((a)^(2)+ (b)^(2))*(a*sin(b*x)- b*cos(b*x)) Integrate[Exp[a*x]*Sin[b*x], x]=Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Sin[b*x]- b*Cos[b*x]) Successful Successful - -
4.26.E8 ∫ e a ⁒ x ⁒ cos ⁑ ( b ⁒ x ) ⁒ d x = e a ⁒ x a 2 + b 2 ⁒ ( a ⁒ cos ⁑ ( b ⁒ x ) + b ⁒ sin ⁑ ( b ⁒ x ) ) superscript 𝑒 π‘Ž π‘₯ 𝑏 π‘₯ π‘₯ superscript 𝑒 π‘Ž π‘₯ superscript π‘Ž 2 superscript 𝑏 2 π‘Ž 𝑏 π‘₯ 𝑏 𝑏 π‘₯ {\displaystyle{\displaystyle\int e^{ax}\cos\left(bx\right)\mathrm{d}x=\frac{e^% {ax}}{a^{2}+b^{2}}(a\cos\left(bx\right)+b\sin\left(bx\right))}} int(exp(a*x)*cos(b*x), x)=(exp(a*x))/((a)^(2)+ (b)^(2))*(a*cos(b*x)+ b*sin(b*x)) Integrate[Exp[a*x]*Cos[b*x], x]=Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Cos[b*x]+ b*Sin[b*x]) Successful Successful - -
4.26.E9 ∫ 0 Ο€ sin ⁑ ( m ⁒ t ) ⁒ sin ⁑ ( n ⁒ t ) ⁒ d t = 0 superscript subscript 0 πœ‹ π‘š 𝑑 𝑛 𝑑 𝑑 0 {\displaystyle{\displaystyle\int_{0}^{\pi}\sin\left(mt\right)\sin\left(nt% \right)\mathrm{d}t=0}} int(sin(m*t)*sin(n*t), t = 0..Pi)= 0 Integrate[Sin[m*t]*Sin[n*t], {t, 0, Pi}]= 0 Failure Failure Skip Successful
4.26.E10 ∫ 0 Ο€ cos ⁑ ( m ⁒ t ) ⁒ cos ⁑ ( n ⁒ t ) ⁒ d t = 0 superscript subscript 0 πœ‹ π‘š 𝑑 𝑛 𝑑 𝑑 0 {\displaystyle{\displaystyle\int_{0}^{\pi}\cos\left(mt\right)\cos\left(nt% \right)\mathrm{d}t=0}} int(cos(m*t)*cos(n*t), t = 0..Pi)= 0 Integrate[Cos[m*t]*Cos[n*t], {t, 0, Pi}]= 0 Failure Failure Skip Successful
4.26.E11 ∫ 0 Ο€ sin 2 ⁑ ( n ⁒ t ) ⁒ d t = ∫ 0 Ο€ cos 2 ⁑ ( n ⁒ t ) ⁒ d t superscript subscript 0 πœ‹ 2 𝑛 𝑑 𝑑 superscript subscript 0 πœ‹ 2 𝑛 𝑑 𝑑 {\displaystyle{\displaystyle\int_{0}^{\pi}{\sin^{2}}\left(nt\right)\mathrm{d}t% =\int_{0}^{\pi}{\cos^{2}}\left(nt\right)\mathrm{d}t}} int((sin(n*t))^(2), t = 0..Pi)= int((cos(n*t))^(2), t = 0..Pi) Integrate[(Sin[n*t])^(2), {t, 0, Pi}]= Integrate[(Cos[n*t])^(2), {t, 0, Pi}] Failure Failure Skip Successful
4.26.E11 ∫ 0 Ο€ cos 2 ⁑ ( n ⁒ t ) ⁒ d t = 1 2 ⁒ Ο€ superscript subscript 0 πœ‹ 2 𝑛 𝑑 𝑑 1 2 πœ‹ {\displaystyle{\displaystyle\int_{0}^{\pi}{\cos^{2}}\left(nt\right)\mathrm{d}t% =\tfrac{1}{2}\pi}} int((cos(n*t))^(2), t = 0..Pi)=(1)/(2)*Pi Integrate[(Cos[n*t])^(2), {t, 0, Pi}]=Divide[1,2]*Pi Failure Failure Skip Successful
4.26.E12 ∫ 0 ∞ sin ⁑ ( m ⁒ t ) t ⁒ d t = { superscript subscript 0 π‘š 𝑑 𝑑 𝑑 cases {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\sin\left(mt\right)}{t}% \mathrm{d}t=\begin{cases}\frac{1}{2}\pi,&m}}\)\@add@PDF@RDFa@triples% \end{document}\end{cases} int((sin(m*t))/(t), t = 0..infinity)= Integrate[Divide[Sin[m*t],t], {t, 0, Infinity}]= Error Failure - Error
4.26.E13 ∫ 0 ∞ sin ⁑ ( t 2 ) ⁒ d t = ∫ 0 ∞ cos ⁑ ( t 2 ) ⁒ d t superscript subscript 0 superscript 𝑑 2 𝑑 superscript subscript 0 superscript 𝑑 2 𝑑 {\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(t^{2}\right)\mathrm{d}t% =\int_{0}^{\infty}\cos\left(t^{2}\right)\mathrm{d}t}} int(sin((t)^(2)), t = 0..infinity)= int(cos((t)^(2)), t = 0..infinity) Integrate[Sin[(t)^(2)], {t, 0, Infinity}]= Integrate[Cos[(t)^(2)], {t, 0, Infinity}] Successful Successful - -
4.26.E13 ∫ 0 ∞ cos ⁑ ( t 2 ) ⁒ d t = 1 2 ⁒ Ο€ 2 superscript subscript 0 superscript 𝑑 2 𝑑 1 2 πœ‹ 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(t^{2}\right)\mathrm{d}t% =\frac{1}{2}\sqrt{\frac{\pi}{2}}}} int(cos((t)^(2)), t = 0..infinity)=(1)/(2)*sqrt((Pi)/(2)) Integrate[Cos[(t)^(2)], {t, 0, Infinity}]=Divide[1,2]*Sqrt[Divide[Pi,2]] Successful Successful - -
4.26.E14 ∫ arcsin ⁑ x ⁒ d x = x ⁒ arcsin ⁑ x + ( 1 - x 2 ) 1 / 2 π‘₯ π‘₯ π‘₯ π‘₯ superscript 1 superscript π‘₯ 2 1 2 {\displaystyle{\displaystyle\int\operatorname{arcsin}x\mathrm{d}x=x% \operatorname{arcsin}x+(1-x^{2})^{1/2}}} int(arcsin(x), x)= x*arcsin(x)+(1 - (x)^(2))^(1/ 2) Integrate[ArcSin[x], x]= x*ArcSin[x]+(1 - (x)^(2))^(1/ 2) Successful Successful - -
4.26.E15 ∫ arccos ⁑ x ⁒ d x = x ⁒ arccos ⁑ x - ( 1 - x 2 ) 1 / 2 π‘₯ π‘₯ π‘₯ π‘₯ superscript 1 superscript π‘₯ 2 1 2 {\displaystyle{\displaystyle\int\operatorname{arccos}x\mathrm{d}x=x% \operatorname{arccos}x-(1-x^{2})^{1/2}}} int(arccos(x), x)= x*arccos(x)-(1 - (x)^(2))^(1/ 2) Integrate[ArcCos[x], x]= x*ArcCos[x]-(1 - (x)^(2))^(1/ 2) Successful Successful - -
4.26.E16 ∫ arctan ⁑ x ⁒ d x = x ⁒ arctan ⁑ x - 1 2 ⁒ ln ⁑ ( 1 + x 2 ) π‘₯ π‘₯ π‘₯ π‘₯ 1 2 1 superscript π‘₯ 2 {\displaystyle{\displaystyle\int\operatorname{arctan}x\mathrm{d}x=x% \operatorname{arctan}x-\tfrac{1}{2}\ln\left(1+x^{2}\right)}} int(arctan(x), x)= x*arctan(x)-(1)/(2)*ln(1 + (x)^(2)) Integrate[ArcTan[x], x]= x*ArcTan[x]-Divide[1,2]*Log[1 + (x)^(2)] Successful Successful - -
4.26.E17 ∫ arccsc ⁑ x ⁒ d x = x ⁒ arccsc ⁑ x + ln ⁑ ( x + ( x 2 - 1 ) 1 / 2 ) π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ superscript superscript π‘₯ 2 1 1 2 {\displaystyle{\displaystyle\int\operatorname{arccsc}x\mathrm{d}x=x% \operatorname{arccsc}x+\ln\left(x+(x^{2}-1)^{1/2}\right)}} int(arccsc(x), x)= x*arccsc(x)+ ln(x +((x)^(2)- 1)^(1/ 2)) Integrate[ArcCsc[x], x]= x*ArcCsc[x]+ Log[x +((x)^(2)- 1)^(1/ 2)] Successful Failure -
Fail
Complex[-4.440892098500626*^-16, -1.5707963267948966] <- {Rule[x, 2]}
Complex[6.661338147750939*^-16, -1.5707963267948966] <- {Rule[x, 3]}
4.26.E18 ∫ arcsec ⁑ x ⁒ d x = x ⁒ arcsec ⁑ x - ln ⁑ ( x + ( x 2 - 1 ) 1 / 2 ) π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ superscript superscript π‘₯ 2 1 1 2 {\displaystyle{\displaystyle\int\operatorname{arcsec}x\mathrm{d}x=x% \operatorname{arcsec}x-\ln\left(x+(x^{2}-1)^{1/2}\right)}} int(arcsec(x), x)= x*arcsec(x)- ln(x +((x)^(2)- 1)^(1/ 2)) Integrate[ArcSec[x], x]= x*ArcSec[x]- Log[x +((x)^(2)- 1)^(1/ 2)] Successful Failure -
Fail
Complex[4.440892098500626*^-16, 1.5707963267948966] <- {Rule[x, 2]}
Complex[-6.661338147750939*^-16, 1.5707963267948966] <- {Rule[x, 3]}
4.26.E19 ∫ arccot ⁑ x ⁒ d x = x ⁒ arccot ⁑ x + 1 2 ⁒ ln ⁑ ( 1 + x 2 ) π‘₯ π‘₯ π‘₯ π‘₯ 1 2 1 superscript π‘₯ 2 {\displaystyle{\displaystyle\int\operatorname{arccot}x\mathrm{d}x=x% \operatorname{arccot}x+\tfrac{1}{2}\ln\left(1+x^{2}\right)}} int(arccot(x), x)= x*arccot(x)+(1)/(2)*ln(1 + (x)^(2)) Integrate[ArcCot[x], x]= x*ArcCot[x]+Divide[1,2]*Log[1 + (x)^(2)] Successful Successful - -
4.26.E20 ∫ x ⁒ arcsin ⁑ x ⁒ d x = ( x 2 2 - 1 4 ) ⁒ arcsin ⁑ x + x 4 ⁒ ( 1 - x 2 ) 1 / 2 π‘₯ π‘₯ π‘₯ superscript π‘₯ 2 2 1 4 π‘₯ π‘₯ 4 superscript 1 superscript π‘₯ 2 1 2 {\displaystyle{\displaystyle\int x\operatorname{arcsin}x\mathrm{d}x=\left(% \frac{x^{2}}{2}-\frac{1}{4}\right)\operatorname{arcsin}x+\frac{x}{4}(1-x^{2})^% {1/2}}} int(x*arcsin(x), x)=(((x)^(2))/(2)-(1)/(4))* arcsin(x)+(x)/(4)*(1 - (x)^(2))^(1/ 2) Integrate[x*ArcSin[x], x]=(Divide[(x)^(2),2]-Divide[1,4])* ArcSin[x]+Divide[x,4]*(1 - (x)^(2))^(1/ 2) Successful Successful - -
4.26.E21 ∫ x ⁒ arccos ⁑ x ⁒ d x = ( x 2 2 - 1 4 ) ⁒ arccos ⁑ x - x 4 ⁒ ( 1 - x 2 ) 1 / 2 π‘₯ π‘₯ π‘₯ superscript π‘₯ 2 2 1 4 π‘₯ π‘₯ 4 superscript 1 superscript π‘₯ 2 1 2 {\displaystyle{\displaystyle\int x\operatorname{arccos}x\mathrm{d}x=\left(% \frac{x^{2}}{2}-\frac{1}{4}\right)\operatorname{arccos}x-\frac{x}{4}(1-x^{2})^% {1/2}}} int(x*arccos(x), x)=(((x)^(2))/(2)-(1)/(4))* arccos(x)-(x)/(4)*(1 - (x)^(2))^(1/ 2) Integrate[x*ArcCos[x], x]=(Divide[(x)^(2),2]-Divide[1,4])* ArcCos[x]-Divide[x,4]*(1 - (x)^(2))^(1/ 2) Failure Failure Skip
Fail
0.39269908169872414 <- {Rule[x, Rational[1, 2]]}
4.28.E1 sinh ⁑ z = e z - e - z 2 𝑧 superscript 𝑒 𝑧 superscript 𝑒 𝑧 2 {\displaystyle{\displaystyle\sinh z=\frac{e^{z}-e^{-z}}{2}}} sinh(z)=(exp(z)- exp(- z))/(2) Sinh[z]=Divide[Exp[z]- Exp[- z],2] Successful Successful - -
4.28.E2 cosh ⁑ z = e z + e - z 2 𝑧 superscript 𝑒 𝑧 superscript 𝑒 𝑧 2 {\displaystyle{\displaystyle\cosh z=\frac{e^{z}+e^{-z}}{2}}} cosh(z)=(exp(z)+ exp(- z))/(2) Cosh[z]=Divide[Exp[z]+ Exp[- z],2] Successful Successful - -
4.28.E3 cosh ⁑ z + sinh ⁑ z = e + z 𝑧 𝑧 superscript 𝑒 𝑧 {\displaystyle{\displaystyle\cosh z+\sinh z=e^{+z}}} cosh(z)+ sinh(z)= exp(+ z) Cosh[z]+ Sinh[z]= Exp[+ z] Successful Successful - -
4.28.E3 cosh ⁑ z - sinh ⁑ z = e - z 𝑧 𝑧 superscript 𝑒 𝑧 {\displaystyle{\displaystyle\cosh z-\sinh z=e^{-z}}} cosh(z)- sinh(z)= exp(- z) Cosh[z]- Sinh[z]= Exp[- z] Successful Successful - -
4.28.E4 tanh ⁑ z = sinh ⁑ z cosh ⁑ z 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\tanh z=\frac{\sinh z}{\cosh z}}} tanh(z)=(sinh(z))/(cosh(z)) Tanh[z]=Divide[Sinh[z],Cosh[z]] Successful Successful - -
4.28.E5 csch ⁑ z = 1 sinh ⁑ z 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{csch}z=\frac{1}{\sinh z}}} csch(z)=(1)/(sinh(z)) Csch[z]=Divide[1,Sinh[z]] Successful Successful - -
4.28.E6 sech ⁑ z = 1 cosh ⁑ z 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{sech}z=\frac{1}{\cosh z}}} sech(z)=(1)/(cosh(z)) Sech[z]=Divide[1,Cosh[z]] Successful Successful - -
4.28.E7 coth ⁑ z = 1 tanh ⁑ z hyperbolic-cotangent 𝑧 1 𝑧 {\displaystyle{\displaystyle\coth z=\frac{1}{\tanh z}}} coth(z)=(1)/(tanh(z)) Coth[z]=Divide[1,Tanh[z]] Successful Successful - -
4.28.E8 sin ⁑ ( i ⁒ z ) = i ⁒ sinh ⁑ z 𝑖 𝑧 𝑖 𝑧 {\displaystyle{\displaystyle\sin\left(iz\right)=i\sinh z}} sin(I*z)= I*sinh(z) Sin[I*z]= I*Sinh[z] Successful Successful - -
4.28.E9 cos ⁑ ( i ⁒ z ) = cosh ⁑ z 𝑖 𝑧 𝑧 {\displaystyle{\displaystyle\cos\left(iz\right)=\cosh z}} cos(I*z)= cosh(z) Cos[I*z]= Cosh[z] Successful Successful - -
4.28.E10 tan ⁑ ( i ⁒ z ) = i ⁒ tanh ⁑ z 𝑖 𝑧 𝑖 𝑧 {\displaystyle{\displaystyle\tan\left(iz\right)=i\tanh z}} tan(I*z)= I*tanh(z) Tan[I*z]= I*Tanh[z] Successful Successful - -
4.28.E11 csc ⁑ ( i ⁒ z ) = - i ⁒ csch ⁑ z 𝑖 𝑧 𝑖 𝑧 {\displaystyle{\displaystyle\csc\left(iz\right)=-i\operatorname{csch}z}} csc(I*z)= - I*csch(z) Csc[I*z]= - I*Csch[z] Successful Successful - -
4.28.E12 sec ⁑ ( i ⁒ z ) = sech ⁑ z 𝑖 𝑧 𝑧 {\displaystyle{\displaystyle\sec\left(iz\right)=\operatorname{sech}z}} sec(I*z)= sech(z) Sec[I*z]= Sech[z] Successful Successful - -
4.28.E13 cot ⁑ ( i ⁒ z ) = - i ⁒ coth ⁑ z 𝑖 𝑧 𝑖 hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\cot\left(iz\right)=-i\coth z}} cot(I*z)= - I*coth(z) Cot[I*z]= - I*Coth[z] Successful Successful - -
4.31.E1 lim z β†’ 0 ⁑ sinh ⁑ z z = 1 subscript β†’ 𝑧 0 𝑧 𝑧 1 {\displaystyle{\displaystyle\lim_{z\to 0}\frac{\sinh z}{z}=1}} limit((sinh(z))/(z), z = 0)= 1 Limit[Divide[Sinh[z],z], z -> 0]= 1 Successful Successful - -
4.31.E2 lim z β†’ 0 ⁑ tanh ⁑ z z = 1 subscript β†’ 𝑧 0 𝑧 𝑧 1 {\displaystyle{\displaystyle\lim_{z\to 0}\frac{\tanh z}{z}=1}} limit((tanh(z))/(z), z = 0)= 1 Limit[Divide[Tanh[z],z], z -> 0]= 1 Successful Successful - -
4.31.E3 lim z β†’ 0 ⁑ cosh ⁑ z - 1 z 2 = 1 2 subscript β†’ 𝑧 0 𝑧 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}}} limit((cosh(z)- 1)/((z)^(2)), z = 0)=(1)/(2) Limit[Divide[Cosh[z]- 1,(z)^(2)], z -> 0]=Divide[1,2] Successful Successful - -
4.32.E1 cosh ⁑ x ≀ ( sinh ⁑ x x ) 3 π‘₯ superscript π‘₯ π‘₯ 3 {\displaystyle{\displaystyle\cosh x<=\left(\frac{\sinh x}{x}\right)^{3}}} cosh(x)< =((sinh(x))/(x))^(3) Cosh[x]< =(Divide[Sinh[x],x])^(3) Failure Failure Successful Successful
4.32.E2 sin ⁑ x ⁒ cos ⁑ x < tanh ⁑ x π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\sin x\cos x<\tanh x}} sin(x)*cos(x)< tanh(x) Sin[x]*Cos[x]< Tanh[x] Failure Failure Successful Successful
4.32.E2 tanh ⁑ x < x π‘₯ π‘₯ {\displaystyle{\displaystyle\tanh x<x}} tanh(x)< x Tanh[x]< x Failure Failure Successful Successful
4.32.E3 | cosh ⁑ x - cosh ⁑ y | β‰₯ | x - y | ⁒ sinh ⁑ x ⁒ sinh ⁑ y π‘₯ 𝑦 π‘₯ 𝑦 π‘₯ 𝑦 {\displaystyle{\displaystyle|\cosh x-\cosh y|>=|x-y|\sqrt{\sinh x\sinh y}}} abs(cosh(x)- cosh(y))> =abs(x - y)*sqrt(sinh(x)*sinh(y)) Abs[Cosh[x]- Cosh[y]]> =Abs[x - y]*Sqrt[Sinh[x]*Sinh[y]] Failure Failure Successful Successful
4.32.E4 arctan ⁑ x ≀ 1 2 ⁒ Ο€ ⁒ tanh ⁑ x π‘₯ 1 2 πœ‹ π‘₯ {\displaystyle{\displaystyle\operatorname{arctan}x<=\tfrac{1}{2}\pi\tanh x}} arctan(x)< =(1)/(2)*Pi*tanh(x) ArcTan[x]< =Divide[1,2]*Pi*Tanh[x] Failure Failure Successful Successful
4.34.E1 d d z ⁑ sinh ⁑ z = cosh ⁑ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sinh z=\cosh z}} diff(sinh(z), z)= cosh(z) D[Sinh[z], z]= Cosh[z] Successful Successful - -
4.34.E2 d d z ⁑ cosh ⁑ z = sinh ⁑ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cosh z=\sinh z}} diff(cosh(z), z)= sinh(z) D[Cosh[z], z]= Sinh[z] Successful Successful - -
4.34.E3 d d z ⁑ tanh ⁑ z = sech 2 ⁑ z derivative 𝑧 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tanh z={% \operatorname{sech}^{2}}z}} diff(tanh(z), z)= (sech(z))^(2) D[Tanh[z], z]= (Sech[z])^(2) Successful Successful - -
4.34.E4 d d z ⁑ csch ⁑ z = - csch ⁑ z ⁒ coth ⁑ z derivative 𝑧 𝑧 𝑧 hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{csch}z% =-\operatorname{csch}z\coth z}} diff(csch(z), z)= - csch(z)*coth(z) D[Csch[z], z]= - Csch[z]*Coth[z] Successful Successful - -
4.34.E5 d d z ⁑ sech ⁑ z = - sech ⁑ z ⁒ tanh ⁑ z derivative 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sech}z% =-\operatorname{sech}z\tanh z}} diff(sech(z), z)= - sech(z)*tanh(z) D[Sech[z], z]= - Sech[z]*Tanh[z] Successful Successful - -
4.34.E6 d d z ⁑ coth ⁑ z = - csch 2 ⁑ z derivative 𝑧 hyperbolic-cotangent 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\coth z=-{% \operatorname{csch}^{2}}z}} diff(coth(z), z)= - (csch(z))^(2) D[Coth[z], z]= - (Csch[z])^(2) Successful Successful - -
4.34.E7 d 2 w d z 2 - a 2 ⁒ w = 0 derivative 𝑀 𝑧 2 superscript π‘Ž 2 𝑀 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w% =0}} diff(w, [z$(2)])- (a)^(2)* w = 0 D[w, {z, 2}]- (a)^(2)* w = 0 Failure Failure
Fail
5.656854245-5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
-5.656854245-5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
-5.656854245+5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
5.656854245+5.656854245*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.34.E8 ( d w d z ) 2 - a 2 ⁒ w 2 = 1 superscript derivative 𝑀 𝑧 2 superscript π‘Ž 2 superscript 𝑀 2 1 {\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a% ^{2}w^{2}=1}} (diff(w, z))^(2)- (a)^(2)* (w)^(2)= 1 (D[w, z])^(2)- (a)^(2)* (w)^(2)= 1 Failure Failure
Fail
14.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
-16.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
14.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-16.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.34.E9 ( d w d z ) 2 - a 2 ⁒ w 2 = - 1 superscript derivative 𝑀 𝑧 2 superscript π‘Ž 2 superscript 𝑀 2 1 {\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a% ^{2}w^{2}=-1}} (diff(w, z))^(2)- (a)^(2)* (w)^(2)= - 1 (D[w, z])^(2)- (a)^(2)* (w)^(2)= - 1 Failure Failure
Fail
16.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
-14.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
16.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-14.99999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
-15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.34.E10 d w d z + a 2 ⁒ w 2 = 1 derivative 𝑀 𝑧 superscript π‘Ž 2 superscript 𝑀 2 1 {\displaystyle{\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}=1}} diff(w, z)+ (a)^(2)* (w)^(2)= 1 D[w, z]+ (a)^(2)* (w)^(2)= 1 Failure Failure
Fail
-16.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
14.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
-16.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
14.99999998+0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.34.E11 w = A ⁒ cosh ⁑ ( a ⁒ z ) + B ⁒ sinh ⁑ ( a ⁒ z ) 𝑀 𝐴 π‘Ž 𝑧 𝐡 π‘Ž 𝑧 {\displaystyle{\displaystyle w=A\cosh\left(az\right)+B\sinh\left(az\right)}} w = A*cosh(a*z)+ B*sinh(a*z) w = A*Cosh[a*z]+ B*Sinh[a*z] Failure Failure Skip Skip
4.34.E12 w = ( 1 / a ) ⁒ sinh ⁑ ( a ⁒ z + c ) 𝑀 1 π‘Ž π‘Ž 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\sinh\left(az+c\right)}} w =(1/ a)* sinh(a*z + c) w =(1/ a)* Sinh[a*z + c] Failure Failure
Fail
1.560620374+2.444010722*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-43.99146068-31.60984586*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
2.401710418+1.239374278*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-.552876601-1.280562047*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.5606203716754656, 2.444010722839117] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-43.991460739515965, -31.609845930171662] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.4017104180648507, 1.2393742767469784] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.5528766038746884, -1.2805620501609194] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.34.E13 w = ( 1 / a ) ⁒ cosh ⁑ ( a ⁒ z + c ) 𝑀 1 π‘Ž π‘Ž 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\cosh\left(az+c\right)}} w =(1/ a)* cosh(a*z + c) w =(1/ a)* Cosh[a*z + c] Failure Failure
Fail
1.439486274+2.433863476*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-43.99015111-31.60804529*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
2.429374695+1.121005682*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
3.350840350+4.086833100*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.4394862722813944, 2.433863476686773] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-43.990151188366596, -31.608045369688465] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.4293746952834034, 1.121005681222388] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.350840354280354, 4.086833103365105] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.34.E14 w = ( 1 / a ) ⁒ coth ⁑ ( a ⁒ z + c ) 𝑀 1 π‘Ž hyperbolic-cotangent π‘Ž 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\coth\left(az+c\right)}} w =(1/ a)* coth(a*z + c) w =(1/ a)* Coth[a*z + c] Failure Failure
Fail
1.029591669+1.718277978*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.060677829+1.767757934*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.083172262+1.824036919*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.765190374+1.065683009*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.0295916694660097, 1.7182779787395532] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.0606778291092909, 1.7677579338582863] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.083172262208037, 1.824036919584044] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.7651903746588453, 1.065683009892447] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.35.E1 sinh ⁑ ( u + v ) = sinh ⁑ u ⁒ cosh ⁑ v + cosh ⁑ u ⁒ sinh ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\sinh\left(u+v\right)=\sinh u\cosh v+\cosh u\sinh v}} sinh(u + v)= sinh(u)*cosh(v)+ cosh(u)*sinh(v) Sinh[u + v]= Sinh[u]*Cosh[v]+ Cosh[u]*Sinh[v] Successful Successful - -
4.35.E1 sinh ⁑ ( u - v ) = sinh ⁑ u ⁒ cosh ⁑ v - cosh ⁑ u ⁒ sinh ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\sinh\left(u-v\right)=\sinh u\cosh v-\cosh u\sinh v}} sinh(u - v)= sinh(u)*cosh(v)- cosh(u)*sinh(v) Sinh[u - v]= Sinh[u]*Cosh[v]- Cosh[u]*Sinh[v] Successful Successful - -
4.35.E2 cosh ⁑ ( u + v ) = cosh ⁑ u ⁒ cosh ⁑ v + sinh ⁑ u ⁒ sinh ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\cosh\left(u+v\right)=\cosh u\cosh v+\sinh u\sinh v}} cosh(u + v)= cosh(u)*cosh(v)+ sinh(u)*sinh(v) Cosh[u + v]= Cosh[u]*Cosh[v]+ Sinh[u]*Sinh[v] Successful Successful - -
4.35.E2 cosh ⁑ ( u - v ) = cosh ⁑ u ⁒ cosh ⁑ v - sinh ⁑ u ⁒ sinh ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\cosh\left(u-v\right)=\cosh u\cosh v-\sinh u\sinh v}} cosh(u - v)= cosh(u)*cosh(v)- sinh(u)*sinh(v) Cosh[u - v]= Cosh[u]*Cosh[v]- Sinh[u]*Sinh[v] Successful Successful - -
4.35.E3 tanh ⁑ ( u + v ) = tanh ⁑ u + tanh ⁑ v 1 + tanh ⁑ u ⁒ tanh ⁑ v 𝑒 𝑣 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\tanh\left(u+v\right)=\frac{\tanh u+\tanh v}{1+% \tanh u\tanh v}}} tanh(u + v)=(tanh(u)+ tanh(v))/(1 + tanh(u)*tanh(v)) Tanh[u + v]=Divide[Tanh[u]+ Tanh[v],1 + Tanh[u]*Tanh[v]] Successful Successful - -
4.35.E3 tanh ⁑ ( u - v ) = tanh ⁑ u - tanh ⁑ v 1 - tanh ⁑ u ⁒ tanh ⁑ v 𝑒 𝑣 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\tanh\left(u-v\right)=\frac{\tanh u-\tanh v}{1-% \tanh u\tanh v}}} tanh(u - v)=(tanh(u)- tanh(v))/(1 - tanh(u)*tanh(v)) Tanh[u - v]=Divide[Tanh[u]- Tanh[v],1 - Tanh[u]*Tanh[v]] Successful Successful - -
4.35.E4 coth ⁑ ( u + v ) = + coth ⁑ u ⁒ coth ⁑ v + 1 coth ⁑ u + coth ⁑ v hyperbolic-cotangent 𝑒 𝑣 hyperbolic-cotangent 𝑒 hyperbolic-cotangent 𝑣 1 hyperbolic-cotangent 𝑒 hyperbolic-cotangent 𝑣 {\displaystyle{\displaystyle\coth\left(u+v\right)=\frac{+\coth u\coth v+1}{% \coth u+\coth v}}} coth(u + v)=(+ coth(u)*coth(v)+ 1)/(coth(u)+ coth(v)) Coth[u + v]=Divide[+ Coth[u]*Coth[v]+ 1,Coth[u]+ Coth[v]] Successful Successful - -
4.35.E4 coth ⁑ ( u - v ) = - coth ⁑ u ⁒ coth ⁑ v + 1 coth ⁑ u - coth ⁑ v hyperbolic-cotangent 𝑒 𝑣 hyperbolic-cotangent 𝑒 hyperbolic-cotangent 𝑣 1 hyperbolic-cotangent 𝑒 hyperbolic-cotangent 𝑣 {\displaystyle{\displaystyle\coth\left(u-v\right)=\frac{-\coth u\coth v+1}{% \coth u-\coth v}}} coth(u - v)=(- coth(u)*coth(v)+ 1)/(coth(u)- coth(v)) Coth[u - v]=Divide[- Coth[u]*Coth[v]+ 1,Coth[u]- Coth[v]] Successful Successful - -
4.35.E5 sinh ⁑ u + sinh ⁑ v = 2 ⁒ sinh ⁑ ( u + v 2 ) ⁒ cosh ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\sinh u+\sinh v=2\sinh\left(\frac{u+v}{2}\right)% \cosh\left(\frac{u-v}{2}\right)}} sinh(u)+ sinh(v)= 2*sinh((u + v)/(2))*cosh((u - v)/(2)) Sinh[u]+ Sinh[v]= 2*Sinh[Divide[u + v,2]]*Cosh[Divide[u - v,2]] Successful Successful - -
4.35.E6 sinh ⁑ u - sinh ⁑ v = 2 ⁒ cosh ⁑ ( u + v 2 ) ⁒ sinh ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\sinh u-\sinh v=2\cosh\left(\frac{u+v}{2}\right)% \sinh\left(\frac{u-v}{2}\right)}} sinh(u)- sinh(v)= 2*cosh((u + v)/(2))*sinh((u - v)/(2)) Sinh[u]- Sinh[v]= 2*Cosh[Divide[u + v,2]]*Sinh[Divide[u - v,2]] Successful Successful - -
4.35.E7 cosh ⁑ u + cosh ⁑ v = 2 ⁒ cosh ⁑ ( u + v 2 ) ⁒ cosh ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\cosh u+\cosh v=2\cosh\left(\frac{u+v}{2}\right)% \cosh\left(\frac{u-v}{2}\right)}} cosh(u)+ cosh(v)= 2*cosh((u + v)/(2))*cosh((u - v)/(2)) Cosh[u]+ Cosh[v]= 2*Cosh[Divide[u + v,2]]*Cosh[Divide[u - v,2]] Successful Successful - -
4.35.E8 cosh ⁑ u - cosh ⁑ v = 2 ⁒ sinh ⁑ ( u + v 2 ) ⁒ sinh ⁑ ( u - v 2 ) 𝑒 𝑣 2 𝑒 𝑣 2 𝑒 𝑣 2 {\displaystyle{\displaystyle\cosh u-\cosh v=2\sinh\left(\frac{u+v}{2}\right)% \sinh\left(\frac{u-v}{2}\right)}} cosh(u)- cosh(v)= 2*sinh((u + v)/(2))*sinh((u - v)/(2)) Cosh[u]- Cosh[v]= 2*Sinh[Divide[u + v,2]]*Sinh[Divide[u - v,2]] Successful Successful - -
4.35.E9 tanh ⁑ u + tanh ⁑ v = sinh ⁑ ( u + v ) cosh ⁑ u ⁒ cosh ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\tanh u+\tanh v=\frac{\sinh\left(u+v\right)}{\cosh u% \cosh v}}} tanh(u)+ tanh(v)=(sinh(u + v))/(cosh(u)*cosh(v)) Tanh[u]+ Tanh[v]=Divide[Sinh[u + v],Cosh[u]*Cosh[v]] Successful Successful - -
4.35.E9 tanh ⁑ u - tanh ⁑ v = sinh ⁑ ( u - v ) cosh ⁑ u ⁒ cosh ⁑ v 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle\tanh u-\tanh v=\frac{\sinh\left(u-v\right)}{\cosh u% \cosh v}}} tanh(u)- tanh(v)=(sinh(u - v))/(cosh(u)*cosh(v)) Tanh[u]- Tanh[v]=Divide[Sinh[u - v],Cosh[u]*Cosh[v]] Successful Successful - -
4.35.E10 coth ⁑ u + coth ⁑ v = sinh ⁑ ( v + u ) sinh ⁑ u ⁒ sinh ⁑ v hyperbolic-cotangent 𝑒 hyperbolic-cotangent 𝑣 𝑣 𝑒 𝑒 𝑣 {\displaystyle{\displaystyle\coth u+\coth v=\frac{\sinh\left(v+u\right)}{\sinh u% \sinh v}}} coth(u)+ coth(v)=(sinh(v + u))/(sinh(u)*sinh(v)) Coth[u]+ Coth[v]=Divide[Sinh[v + u],Sinh[u]*Sinh[v]] Successful Successful - -
4.35.E10 coth ⁑ u - coth ⁑ v = sinh ⁑ ( v - u ) sinh ⁑ u ⁒ sinh ⁑ v hyperbolic-cotangent 𝑒 hyperbolic-cotangent 𝑣 𝑣 𝑒 𝑒 𝑣 {\displaystyle{\displaystyle\coth u-\coth v=\frac{\sinh\left(v-u\right)}{\sinh u% \sinh v}}} coth(u)- coth(v)=(sinh(v - u))/(sinh(u)*sinh(v)) Coth[u]- Coth[v]=Divide[Sinh[v - u],Sinh[u]*Sinh[v]] Successful Successful - -
4.35.E11 cosh 2 ⁑ z - sinh 2 ⁑ z = 1 2 𝑧 2 𝑧 1 {\displaystyle{\displaystyle{\cosh^{2}}z-{\sinh^{2}}z=1}} (cosh(z))^(2)- (sinh(z))^(2)= 1 (Cosh[z])^(2)- (Sinh[z])^(2)= 1 Successful Successful - -
4.35.E12 sech 2 ⁑ z = 1 - tanh 2 ⁑ z 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle{\operatorname{sech}^{2}}z=1-{\tanh^{2}}z}} (sech(z))^(2)= 1 - (tanh(z))^(2) (Sech[z])^(2)= 1 - (Tanh[z])^(2) Successful Successful - -
4.35.E13 csch 2 ⁑ z = coth 2 ⁑ z - 1 2 𝑧 hyperbolic-cotangent 2 𝑧 1 {\displaystyle{\displaystyle{\operatorname{csch}^{2}}z={\coth^{2}}z-1}} (csch(z))^(2)= (coth(z))^(2)- 1 (Csch[z])^(2)= (Coth[z])^(2)- 1 Successful Successful - -
4.35.E14 2 ⁒ sinh ⁑ u ⁒ sinh ⁑ v = cosh ⁑ ( u + v ) - cosh ⁑ ( u - v ) 2 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle 2\sinh u\sinh v=\cosh\left(u+v\right)-\cosh\left(% u-v\right)}} 2*sinh(u)*sinh(v)= cosh(u + v)- cosh(u - v) 2*Sinh[u]*Sinh[v]= Cosh[u + v]- Cosh[u - v] Successful Successful - -
4.35.E15 2 ⁒ cosh ⁑ u ⁒ cosh ⁑ v = cosh ⁑ ( u + v ) + cosh ⁑ ( u - v ) 2 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle 2\cosh u\cosh v=\cosh\left(u+v\right)+\cosh\left(% u-v\right)}} 2*cosh(u)*cosh(v)= cosh(u + v)+ cosh(u - v) 2*Cosh[u]*Cosh[v]= Cosh[u + v]+ Cosh[u - v] Successful Successful - -
4.35.E16 2 ⁒ sinh ⁑ u ⁒ cosh ⁑ v = sinh ⁑ ( u + v ) + sinh ⁑ ( u - v ) 2 𝑒 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle 2\sinh u\cosh v=\sinh\left(u+v\right)+\sinh\left(% u-v\right)}} 2*sinh(u)*cosh(v)= sinh(u + v)+ sinh(u - v) 2*Sinh[u]*Cosh[v]= Sinh[u + v]+ Sinh[u - v] Successful Successful - -
4.35.E17 sinh 2 ⁑ u - sinh 2 ⁑ v = sinh ⁑ ( u + v ) ⁒ sinh ⁑ ( u - v ) 2 𝑒 2 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle{\sinh^{2}}u-{\sinh^{2}}v=\sinh\left(u+v\right)% \sinh\left(u-v\right)}} (sinh(u))^(2)- (sinh(v))^(2)= sinh(u + v)*sinh(u - v) (Sinh[u])^(2)- (Sinh[v])^(2)= Sinh[u + v]*Sinh[u - v] Successful Successful - -
4.35.E18 cosh 2 ⁑ u - cosh 2 ⁑ v = sinh ⁑ ( u + v ) ⁒ sinh ⁑ ( u - v ) 2 𝑒 2 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle{\cosh^{2}}u-{\cosh^{2}}v=\sinh\left(u+v\right)% \sinh\left(u-v\right)}} (cosh(u))^(2)- (cosh(v))^(2)= sinh(u + v)*sinh(u - v) (Cosh[u])^(2)- (Cosh[v])^(2)= Sinh[u + v]*Sinh[u - v] Successful Successful - -
4.35.E19 sinh 2 ⁑ u + cosh 2 ⁑ v = cosh ⁑ ( u + v ) ⁒ cosh ⁑ ( u - v ) 2 𝑒 2 𝑣 𝑒 𝑣 𝑒 𝑣 {\displaystyle{\displaystyle{\sinh^{2}}u+{\cosh^{2}}v=\cosh\left(u+v\right)% \cosh\left(u-v\right)}} (sinh(u))^(2)+ (cosh(v))^(2)= cosh(u + v)*cosh(u - v) (Sinh[u])^(2)+ (Cosh[v])^(2)= Cosh[u + v]*Cosh[u - v] Successful Successful - -
4.35.E20 sinh ⁑ z 2 = ( cosh ⁑ z - 1 2 ) 1 / 2 𝑧 2 superscript 𝑧 1 2 1 2 {\displaystyle{\displaystyle\sinh\frac{z}{2}=\left(\frac{\cosh z-1}{2}\right)^% {1/2}}} sinh((z)/(2))=((cosh(z)- 1)/(2))^(1/ 2) Sinh[Divide[z,2]]=(Divide[Cosh[z]- 1,2])^(1/ 2) Failure Failure
Fail
-1.167010648-1.637854044*I <- {z = -2^(1/2)-I*2^(1/2)}
-1.167010648+1.637854044*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-1.1670106484252494, -1.6378540442140963] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.1670106484252494, 1.6378540442140963] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.35.E21 cosh ⁑ z 2 = ( cosh ⁑ z + 1 2 ) 1 / 2 𝑧 2 superscript 𝑧 1 2 1 2 {\displaystyle{\displaystyle\cosh\frac{z}{2}=\left(\frac{\cosh z+1}{2}\right)^% {1/2}}} cosh((z)/(2))=((cosh(z)+ 1)/(2))^(1/ 2) Cosh[Divide[z,2]]=(Divide[Cosh[z]+ 1,2])^(1/ 2) Failure Failure Successful Successful
4.35.E22 tanh ⁑ z 2 = ( cosh ⁑ z - 1 cosh ⁑ z + 1 ) 1 / 2 𝑧 2 superscript 𝑧 1 𝑧 1 1 2 {\displaystyle{\displaystyle\tanh\frac{z}{2}=\left(\frac{\cosh z-1}{\cosh z+1}% \right)^{1/2}}} tanh((z)/(2))=((cosh(z)- 1)/(cosh(z)+ 1))^(1/ 2) Tanh[Divide[z,2]]=(Divide[Cosh[z]- 1,Cosh[z]+ 1])^(1/ 2) Failure Failure
Fail
-1.658064547-.8463685478*I <- {z = -2^(1/2)-I*2^(1/2)}
-1.658064547+.8463685478*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-1.6580645472823399, -0.8463685477398917] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6580645472823399, 0.8463685477398917] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.35.E22 ( cosh ⁑ z - 1 cosh ⁑ z + 1 ) 1 / 2 = cosh ⁑ z - 1 sinh ⁑ z superscript 𝑧 1 𝑧 1 1 2 𝑧 1 𝑧 {\displaystyle{\displaystyle\left(\frac{\cosh z-1}{\cosh z+1}\right)^{1/2}=% \frac{\cosh z-1}{\sinh z}}} ((cosh(z)- 1)/(cosh(z)+ 1))^(1/ 2)=(cosh(z)- 1)/(sinh(z)) (Divide[Cosh[z]- 1,Cosh[z]+ 1])^(1/ 2)=Divide[Cosh[z]- 1,Sinh[z]] Failure Failure
Fail
1.658064547+.8463685479*I <- {z = -2^(1/2)-I*2^(1/2)}
1.658064547-.8463685479*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.6580645472823403, 0.8463685477398917] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.6580645472823403, -0.8463685477398917] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
4.35.E22 cosh ⁑ z - 1 sinh ⁑ z = sinh ⁑ z cosh ⁑ z + 1 𝑧 1 𝑧 𝑧 𝑧 1 {\displaystyle{\displaystyle\frac{\cosh z-1}{\sinh z}=\frac{\sinh z}{\cosh z+1% }}} (cosh(z)- 1)/(sinh(z))=(sinh(z))/(cosh(z)+ 1) Divide[Cosh[z]- 1,Sinh[z]]=Divide[Sinh[z],Cosh[z]+ 1] Successful Successful - -
4.35.E23 sinh ⁑ ( - z ) = - sinh ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\sinh\left(-z\right)=-\sinh z}} sinh(- z)= - sinh(z) Sinh[- z]= - Sinh[z] Successful Successful - -
4.35.E24 cosh ⁑ ( - z ) = cosh ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\cosh\left(-z\right)=\cosh z}} cosh(- z)= cosh(z) Cosh[- z]= Cosh[z] Successful Successful - -
4.35.E25 tanh ⁑ ( - z ) = - tanh ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\tanh\left(-z\right)=-\tanh z}} tanh(- z)= - tanh(z) Tanh[- z]= - Tanh[z] Successful Successful - -
4.35.E26 sinh ⁑ ( 2 ⁒ z ) = 2 ⁒ sinh ⁑ z ⁒ cosh ⁑ z 2 𝑧 2 𝑧 𝑧 {\displaystyle{\displaystyle\sinh\left(2z\right)=2\sinh z\cosh z}} sinh(2*z)= 2*sinh(z)*cosh(z) Sinh[2*z]= 2*Sinh[z]*Cosh[z] Successful Successful - -
4.35.E26 2 ⁒ sinh ⁑ z ⁒ cosh ⁑ z = 2 ⁒ tanh ⁑ z 1 - tanh 2 ⁑ z 2 𝑧 𝑧 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle 2\sinh z\cosh z=\frac{2\tanh z}{1-{\tanh^{2}}z}}} 2*sinh(z)*cosh(z)=(2*tanh(z))/(1 - (tanh(z))^(2)) 2*Sinh[z]*Cosh[z]=Divide[2*Tanh[z],1 - (Tanh[z])^(2)] Successful Successful - -
4.35.E27 cosh ⁑ ( 2 ⁒ z ) = 2 ⁒ cosh 2 ⁑ z - 1 2 𝑧 2 2 𝑧 1 {\displaystyle{\displaystyle\cosh\left(2z\right)=2{\cosh^{2}}z-1}} cosh(2*z)= 2*(cosh(z))^(2)- 1 Cosh[2*z]= 2*(Cosh[z])^(2)- 1 Successful Successful - -
4.35.E27 2 ⁒ sinh 2 ⁑ z + 1 = cosh 2 ⁑ z + sinh 2 ⁑ z 2 2 𝑧 1 2 𝑧 2 𝑧 {\displaystyle{\displaystyle 2{\sinh^{2}}z+1\\ ={\cosh^{2}}z+{\sinh^{2}}z}} 2*(sinh(z))^(2)+ 1 = (cosh(z))^(2)+ (sinh(z))^(2) 2*(Sinh[z])^(2)+ 1 = (Cosh[z])^(2)+ (Sinh[z])^(2) Error Error - -
4.35.E28 tanh ⁑ ( 2 ⁒ z ) = 2 ⁒ tanh ⁑ z 1 + tanh 2 ⁑ z 2 𝑧 2 𝑧 1 2 𝑧 {\displaystyle{\displaystyle\tanh\left(2z\right)=\frac{2\tanh z}{1+{\tanh^{2}}% z}}} tanh(2*z)=(2*tanh(z))/(1 + (tanh(z))^(2)) Tanh[2*z]=Divide[2*Tanh[z],1 + (Tanh[z])^(2)] Successful Successful - -
4.35.E29 sinh ⁑ ( 3 ⁒ z ) = 3 ⁒ sinh ⁑ z + 4 ⁒ sinh 3 ⁑ z 3 𝑧 3 𝑧 4 3 𝑧 {\displaystyle{\displaystyle\sinh\left(3z\right)=3\sinh z+4{\sinh^{3}}z}} sinh(3*z)= 3*sinh(z)+ 4*(sinh(z))^(3) Sinh[3*z]= 3*Sinh[z]+ 4*(Sinh[z])^(3) Successful Successful - -
4.35.E30 cosh ⁑ ( 3 ⁒ z ) = - 3 ⁒ cosh ⁑ z + 4 ⁒ cosh 3 ⁑ z 3 𝑧 3 𝑧 4 3 𝑧 {\displaystyle{\displaystyle\cosh\left(3z\right)=-3\cosh z+4{\cosh^{3}}z}} cosh(3*z)= - 3*cosh(z)+ 4*(cosh(z))^(3) Cosh[3*z]= - 3*Cosh[z]+ 4*(Cosh[z])^(3) Successful Successful - -
4.35.E31 sinh ⁑ ( 4 ⁒ z ) = 4 ⁒ sinh 3 ⁑ z ⁒ cosh ⁑ z + 4 ⁒ cosh 3 ⁑ z ⁒ sinh ⁑ z 4 𝑧 4 3 𝑧 𝑧 4 3 𝑧 𝑧 {\displaystyle{\displaystyle\sinh\left(4z\right)=4{\sinh^{3}}z\cosh z+4{\cosh^% {3}}z\sinh z}} sinh(4*z)= 4*(sinh(z))^(3)* cosh(z)+ 4*(cosh(z))^(3)* sinh(z) Sinh[4*z]= 4*(Sinh[z])^(3)* Cosh[z]+ 4*(Cosh[z])^(3)* Sinh[z] Successful Successful - -
4.35.E32 cosh ⁑ ( 4 ⁒ z ) = cosh 4 ⁑ z + 6 ⁒ sinh 2 ⁑ z ⁒ cosh 2 ⁑ z + sinh 4 ⁑ z 4 𝑧 4 𝑧 6 2 𝑧 2 𝑧 4 𝑧 {\displaystyle{\displaystyle\cosh\left(4z\right)={\cosh^{4}}z+6{\sinh^{2}}z{% \cosh^{2}}z+{\sinh^{4}}z}} cosh(4*z)= (cosh(z))^(4)+ 6*(sinh(z))^(2)* (cosh(z))^(2)+ (sinh(z))^(4) Cosh[4*z]= (Cosh[z])^(4)+ 6*(Sinh[z])^(2)* (Cosh[z])^(2)+ (Sinh[z])^(4) Successful Successful - -
4.35.E33 cosh ⁑ ( n ⁒ z ) + sinh ⁑ ( n ⁒ z ) = ( cosh ⁑ z + sinh ⁑ z ) n 𝑛 𝑧 𝑛 𝑧 superscript 𝑧 𝑧 𝑛 {\displaystyle{\displaystyle\cosh\left(nz\right)+\sinh\left(nz\right)=(\cosh z% +\sinh z)^{n}}} cosh(n*z)+ sinh(n*z)=(cosh(z)+ sinh(z))^(n) Cosh[n*z]+ Sinh[n*z]=(Cosh[z]+ Sinh[z])^(n) Successful Failure - Successful
4.35.E33 cosh ⁑ ( n ⁒ z ) - sinh ⁑ ( n ⁒ z ) = ( cosh ⁑ z - sinh ⁑ z ) n 𝑛 𝑧 𝑛 𝑧 superscript 𝑧 𝑧 𝑛 {\displaystyle{\displaystyle\cosh\left(nz\right)-\sinh\left(nz\right)=(\cosh z% -\sinh z)^{n}}} cosh(n*z)- sinh(n*z)=(cosh(z)- sinh(z))^(n) Cosh[n*z]- Sinh[n*z]=(Cosh[z]- Sinh[z])^(n) Successful Failure - Successful
4.35.E34 sinh ⁑ z = sinh ⁑ x ⁒ cos ⁑ y + i ⁒ cosh ⁑ x ⁒ sin ⁑ y 𝑧 π‘₯ 𝑦 𝑖 π‘₯ 𝑦 {\displaystyle{\displaystyle\sinh z=\sinh x\cos y+i\cosh x\sin y}} sinh(z)= sinh(x)*cos(y)+ I*cosh(x)*sin(y) Sinh[z]= Sinh[x]*Cos[y]+ I*Cosh[x]*Sin[y] Failure Failure
Fail
-.3332024445+.853077958*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
.7908177296+.748416289*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
1.465201834+1.933775988*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-1.657839572-1.014242973*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[-0.33320244491697526, 0.8530779599233089] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.7908177289090546, 0.7484162907172458] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4652018335710113, 1.933775989717134] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.6578395715538454, -1.014242971876882] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.35.E35 cosh ⁑ z = cosh ⁑ x ⁒ cos ⁑ y + i ⁒ sinh ⁑ x ⁒ sin ⁑ y 𝑧 π‘₯ 𝑦 𝑖 π‘₯ 𝑦 {\displaystyle{\displaystyle\cosh z=\cosh x\cos y+i\sinh x\sin y}} cosh(z)= cosh(x)*cos(y)+ I*sinh(x)*sin(y) Cosh[z]= Cosh[x]*Cos[y]+ I*Sinh[x]*Sin[y] Failure Failure
Fail
-.4940560329+.9224954029*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
.9818221171+.842785687*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
1.867312242+1.745548707*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-1.693049015-1.140504690*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[-0.49405603343642457, 0.9224954044013453] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.9818221164102445, 0.842785688781432] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.867312241811268, 1.7455487082452315] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.6930490153249411, -1.1405046889875898] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.35.E36 tanh ⁑ z = sinh ⁑ ( 2 ⁒ x ) + i ⁒ sin ⁑ ( 2 ⁒ y ) cosh ⁑ ( 2 ⁒ x ) + cos ⁑ ( 2 ⁒ y ) 𝑧 2 π‘₯ 𝑖 2 𝑦 2 π‘₯ 2 𝑦 {\displaystyle{\displaystyle\tanh z=\frac{\sinh\left(2x\right)+i\sin\left(2y% \right)}{\cosh\left(2x\right)+\cos\left(2y\right)}}} tanh(z)=(sinh(2*x)+ I*sin(2*y))/(cosh(2*x)+ cos(2*y)) Tanh[z]=Divide[Sinh[2*x]+ I*Sin[2*y],Cosh[2*x]+ Cos[2*y]] Failure Failure
Fail
.34450810e-1-.2308812766*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-.48362120e-1+.2843295099*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.3503564896+.1000398483*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.103580521+.705848261e-2*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[0.03445081006213968, -0.23088127675619197] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.04836211984008565, 0.28432950974904503] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.3503564901139231, 0.1000398481293705] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.1035805212542007, 0.007058482483423091] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.35.E37 coth ⁑ z = sinh ⁑ ( 2 ⁒ x ) - i ⁒ sin ⁑ ( 2 ⁒ y ) cosh ⁑ ( 2 ⁒ x ) - cos ⁑ ( 2 ⁒ y ) hyperbolic-cotangent 𝑧 2 π‘₯ 𝑖 2 𝑦 2 π‘₯ 2 𝑦 {\displaystyle{\displaystyle\coth z=\frac{\sinh\left(2x\right)-i\sin\left(2y% \right)}{\cosh\left(2x\right)-\cos\left(2y\right)}}} coth(z)=(sinh(2*x)- I*sin(2*y))/(cosh(2*x)- cos(2*y)) Coth[z]=Divide[Sinh[2*x]- I*Sin[2*y],Cosh[2*x]- Cos[2*y]] Failure Failure
Fail
.249484083e-1+.1849879852*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
.716327538e-1-.2040171895*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
-.4014084428-.1323526933*I <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-.913666750e-1+.16417892e-3*I <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
Complex[0.024948408389711685, 0.18498798535387256] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.07163275379178491, -0.20401718940971514] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.401408442677776, -0.1323526931640852] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.09136667517255426, 1.6417903322245991*^-4] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.35.E38 | sinh ⁑ z | = ( sinh 2 ⁑ x + sin 2 ⁑ y ) 1 / 2 𝑧 superscript 2 π‘₯ 2 𝑦 1 2 {\displaystyle{\displaystyle|\sinh z|=({\sinh^{2}}x+{\sin^{2}}y)^{1/2}}} abs(sinh(z))=((sinh(x))^(2)+ (sin(y))^(2))^(1/ 2) Abs[Sinh[z]]=((Sinh[x])^(2)+ (Sin[y])^(2))^(1/ 2) Failure Failure
Fail
.727197533 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
.686687095 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.988950286 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-1.550602076 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
0.7271975341555692 <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.6866870962353082 <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.9889502867617381 <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-1.5506020747613944 <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.35.E38 ( sinh 2 ⁑ x + sin 2 ⁑ y ) 1 / 2 = ( 1 2 ⁒ ( cosh ⁑ ( 2 ⁒ x ) - cos ⁑ ( 2 ⁒ y ) ) ) 1 / 2 superscript 2 π‘₯ 2 𝑦 1 2 superscript 1 2 2 π‘₯ 2 𝑦 1 2 {\displaystyle{\displaystyle({\sinh^{2}}x+{\sin^{2}}y)^{1/2}=\left(\tfrac{1}{2% }(\cosh\left(2x\right)-\cos\left(2y\right))\right)^{1/2}}} ((sinh(x))^(2)+ (sin(y))^(2))^(1/ 2)=((1)/(2)*(cosh(2*x)- cos(2*y)))^(1/ 2) ((Sinh[x])^(2)+ (Sin[y])^(2))^(1/ 2)=(Divide[1,2]*(Cosh[2*x]- Cos[2*y]))^(1/ 2) Successful Successful - -
4.35.E39 | cosh ⁑ z | = ( sinh 2 ⁑ x + cos 2 ⁑ y ) 1 / 2 𝑧 superscript 2 π‘₯ 2 𝑦 1 2 {\displaystyle{\displaystyle|\cosh z|=({\sinh^{2}}x+{\cos^{2}}y)^{1/2}}} abs(cosh(z))=((sinh(x))^(2)+ (cos(y))^(2))^(1/ 2) Abs[Cosh[z]]=((Sinh[x])^(2)+ (Cos[y])^(2))^(1/ 2) Failure Failure
Fail
.647885813 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
.694634194 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.404726137 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-1.725544377 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
0.6478858145183544 <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.694634195495673 <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.40472613814456393 <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-1.7255443754392632 <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.35.E39 ( sinh 2 ⁑ x + cos 2 ⁑ y ) 1 / 2 = ( 1 2 ⁒ ( cosh ⁑ ( 2 ⁒ x ) + cos ⁑ ( 2 ⁒ y ) ) ) 1 / 2 superscript 2 π‘₯ 2 𝑦 1 2 superscript 1 2 2 π‘₯ 2 𝑦 1 2 {\displaystyle{\displaystyle({\sinh^{2}}x+{\cos^{2}}y)^{1/2}=\left(\tfrac{1}{2% }(\cosh\left(2x\right)+\cos\left(2y\right))\right)^{1/2}}} ((sinh(x))^(2)+ (cos(y))^(2))^(1/ 2)=((1)/(2)*(cosh(2*x)+ cos(2*y)))^(1/ 2) ((Sinh[x])^(2)+ (Cos[y])^(2))^(1/ 2)=(Divide[1,2]*(Cosh[2*x]+ Cos[2*y]))^(1/ 2) Successful Successful - -
4.35.E40 | tanh ⁑ z | = ( cosh ⁑ ( 2 ⁒ x ) - cos ⁑ ( 2 ⁒ y ) cosh ⁑ ( 2 ⁒ x ) + cos ⁑ ( 2 ⁒ y ) ) 1 / 2 𝑧 superscript 2 π‘₯ 2 𝑦 2 π‘₯ 2 𝑦 1 2 {\displaystyle{\displaystyle|\tanh z|=\left(\frac{\cosh\left(2x\right)-\cos% \left(2y\right)}{\cosh\left(2x\right)+\cos\left(2y\right)}\right)^{1/2}}} abs(tanh(z))=((cosh(2*x)- cos(2*y))/(cosh(2*x)+ cos(2*y)))^(1/ 2) Abs[Tanh[z]]=(Divide[Cosh[2*x]- Cos[2*y],Cosh[2*x]+ Cos[2*y]])^(1/ 2) Failure Failure
Fail
.1650695e-2 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-.72745631e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.3488272508 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.103763936 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
Fail
0.0016506944407532753 <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-0.07274563209398122 <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.3488272498892625 <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.10376393520222194 <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.36.E1 sinh ⁑ z = z ⁒ ∏ n = 1 ∞ ( 1 + z 2 n 2 ⁒ Ο€ 2 ) 𝑧 𝑧 superscript subscript product 𝑛 1 1 superscript 𝑧 2 superscript 𝑛 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\sinh z=z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{% n^{2}\pi^{2}}\right)}} sinh(z)= z*product(1 +((z)^(2))/((n)^(2)* (Pi)^(2)), n = 1..infinity) Sinh[z]= z*Product[1 +Divide[(z)^(2),(n)^(2)* (Pi)^(2)], {n, 1, Infinity}] Failure Successful Skip -
4.36.E2 cosh ⁑ z = ∏ n = 1 ∞ ( 1 + 4 ⁒ z 2 ( 2 ⁒ n - 1 ) 2 ⁒ Ο€ 2 ) 𝑧 superscript subscript product 𝑛 1 1 4 superscript 𝑧 2 superscript 2 𝑛 1 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\cosh z=\prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{% (2n-1)^{2}\pi^{2}}\right)}} cosh(z)= product(1 +(4*(z)^(2))/((2*n - 1)^(2)* (Pi)^(2)), n = 1..infinity) Cosh[z]= Product[1 +Divide[4*(z)^(2),(2*n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}] Failure Successful Skip -
4.36.E3 coth ⁑ z = 1 z + 2 ⁒ z ⁒ βˆ‘ n = 1 ∞ 1 z 2 + n 2 ⁒ Ο€ 2 hyperbolic-cotangent 𝑧 1 𝑧 2 𝑧 superscript subscript 𝑛 1 1 superscript 𝑧 2 superscript 𝑛 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\coth z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{% z^{2}+n^{2}\pi^{2}}}} coth(z)=(1)/(z)+ 2*z*sum((1)/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity) Coth[z]=Divide[1,z]+ 2*z*Sum[Divide[1,(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}] Successful Successful - -
4.36.E4 csch 2 ⁑ z = βˆ‘ n = - ∞ ∞ 1 ( z - n ⁒ Ο€ ⁒ i ) 2 2 𝑧 superscript subscript 𝑛 1 superscript 𝑧 𝑛 πœ‹ 𝑖 2 {\displaystyle{\displaystyle{\operatorname{csch}^{2}}z=\sum_{n=-\infty}^{% \infty}\frac{1}{(z-n\pi i)^{2}}}} (csch(z))^(2)= sum((1)/((z - n*Pi*I)^(2)), n = - infinity..infinity) (Csch[z])^(2)= Sum[Divide[1,(z - n*Pi*I)^(2)], {n, - Infinity, Infinity}] Successful Successful - -
4.36.E5 csch ⁑ z = 1 z + 2 ⁒ z ⁒ βˆ‘ n = 1 ∞ ( - 1 ) n z 2 + n 2 ⁒ Ο€ 2 𝑧 1 𝑧 2 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑧 2 superscript 𝑛 2 superscript πœ‹ 2 {\displaystyle{\displaystyle\operatorname{csch}z=\frac{1}{z}+2z\sum_{n=1}^{% \infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}}} csch(z)=(1)/(z)+ 2*z*sum(((- 1)^(n))/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity) Csch[z]=Divide[1,z]+ 2*z*Sum[Divide[(- 1)^(n),(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}] Successful Successful - -
4.37.E1 Arcsinh ⁑ z = ∫ 0 z d t ( 1 + t 2 ) 1 / 2 multivalued-hyperbolic-inverse-sine 𝑧 superscript subscript 0 𝑧 𝑑 superscript 1 superscript 𝑑 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}z=\int_{0}^{z}\frac{\mathrm{% d}t}{(1+t^{2})^{1/2}}}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, z}]= Integrate[Divide[1,(1 + (t)^(2))^(1/ 2)], {t, 0, z}] Error Failure - Successful
4.37.E2 Arccosh ⁑ z = ∫ 1 z d t ( t 2 - 1 ) 1 / 2 multivalued-hyperbolic-inverse-cosine 𝑧 superscript subscript 1 𝑧 𝑑 superscript superscript 𝑑 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arccosh}z=\int_{1}^{z}\frac{\mathrm{% d}t}{(t^{2}-1)^{1/2}}}} Error Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, z}]= Integrate[Divide[1,((t)^(2)- 1)^(1/ 2)], {t, 1, z}] Error Failure - Error
4.37.E3 Arctanh ⁑ z = ∫ 0 z d t 1 - t 2 multivalued-hyperbolic-inverse-tangent 𝑧 superscript subscript 0 𝑧 𝑑 1 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{Arctanh}z=\int_{0}^{z}\frac{\mathrm{% d}t}{1-t^{2}}}} Error Integrate[Divide[1, 1-t^2], {t, 0, z}]= Integrate[Divide[1,1 - (t)^(2)], {t, 0, z}] Error Failure - Successful
4.37.E4 Arccsch ⁑ z = Arcsinh ⁑ ( 1 / z ) multivalued-hyperbolic-inverse-cosecant 𝑧 multivalued-hyperbolic-inverse-sine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccsch}z=\operatorname{Arcsinh}% \left(1/z\right)}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, Divide[1,z]}]= Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, 1/ z}] Error Failure - Successful
4.37.E5 Arcsech ⁑ z = Arccosh ⁑ ( 1 / z ) multivalued-hyperbolic-inverse-secant 𝑧 multivalued-hyperbolic-inverse-cosine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arcsech}z=\operatorname{Arccosh}% \left(1/z\right)}} Error Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, Divide[1,z]}]= Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, 1/ z}] Error Failure - Error
4.37.E6 Arccoth ⁑ z = Arctanh ⁑ ( 1 / z ) multivalued-hyperbolic-inverse-cotangent 𝑧 multivalued-hyperbolic-inverse-tangent 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccoth}z=\operatorname{Arctanh}% \left(1/z\right)}} Error Integrate[Divide[1, 1-t^2], {t, 0, Divide[1,z]}]= Integrate[Divide[1, 1-t^2], {t, 0, 1/ z}] Error Failure - Successful
4.37.E7 arccsch ⁑ z = arcsinh ⁑ ( 1 / z ) hyperbolic-inverse-cosecant 𝑧 hyperbolic-inverse-sine 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccsch}z=\operatorname{arcsinh}% \left(1/z\right)}} arccsch(z)= arcsinh(1/ z) ArcCsch[z]= ArcSinh[1/ z] Failure Successful Successful -
4.37.E8 arcsech ⁑ z = arccosh ⁑ ( 1 / z ) hyperbolic-inverse-secant 𝑧 hyperbolic-inverse-cosine 1 𝑧 {\displaystyle{\displaystyle\operatorname{arcsech}z=\operatorname{arccosh}% \left(1/z\right)}} arcsech(z)= arccosh(1/ z) ArcSech[z]= ArcCosh[1/ z] Failure Successful Successful -
4.37.E9 arccoth ⁑ z = arctanh ⁑ ( 1 / z ) hyperbolic-inverse-cotangent 𝑧 hyperbolic-inverse-tangent 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccoth}z=\operatorname{arctanh}% \left(1/z\right)}} arccoth(z)= arctanh(1/ z) ArcCoth[z]= ArcTanh[1/ z] Failure Successful Successful -
4.37.E10 arcsinh ⁑ ( - z ) = - arcsinh ⁑ z hyperbolic-inverse-sine 𝑧 hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(-z\right)=-% \operatorname{arcsinh}z}} arcsinh(- z)= - arcsinh(z) ArcSinh[- z]= - ArcSinh[z] Successful Successful - -
4.37.E11 arccosh ⁑ ( - z ) = + Ο€ ⁒ i + arccosh ⁑ z hyperbolic-inverse-cosine 𝑧 πœ‹ 𝑖 hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\left(-z\right)=+\pi i+% \operatorname{arccosh}z}} arccosh(- z)= + Pi*I + arccosh(z) ArcCosh[- z]= + Pi*I + ArcCosh[z] Failure Failure Error Error
4.37.E11 arccosh ⁑ ( - z ) = - Ο€ ⁒ i + arccosh ⁑ z hyperbolic-inverse-cosine 𝑧 πœ‹ 𝑖 hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\left(-z\right)=-\pi i+% \operatorname{arccosh}z}} arccosh(- z)= - Pi*I + arccosh(z) ArcCosh[- z]= - Pi*I + ArcCosh[z] Failure Failure Error Error
4.37.E12 arctanh ⁑ ( - z ) = - arctanh ⁑ z hyperbolic-inverse-tangent 𝑧 hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\left(-z\right)=-% \operatorname{arctanh}z}} arctanh(- z)= - arctanh(z) ArcTanh[- z]= - ArcTanh[z] Successful Successful - -
4.37.E13 arccsch ⁑ ( - z ) = - arccsch ⁑ z hyperbolic-inverse-cosecant 𝑧 hyperbolic-inverse-cosecant 𝑧 {\displaystyle{\displaystyle\operatorname{arccsch}\left(-z\right)=-% \operatorname{arccsch}z}} arccsch(- z)= - arccsch(z) ArcCsch[- z]= - ArcCsch[z] Successful Successful - -
4.37.E14 arcsech ⁑ ( - z ) = - Ο€ ⁒ i + arcsech ⁑ z hyperbolic-inverse-secant 𝑧 πœ‹ 𝑖 hyperbolic-inverse-secant 𝑧 {\displaystyle{\displaystyle\operatorname{arcsech}\left(-z\right)=-\pi i+% \operatorname{arcsech}z}} arcsech(- z)= - Pi*I + arcsech(z) ArcSech[- z]= - Pi*I + ArcSech[z] Failure Failure Error Error
4.37.E14 arcsech ⁑ ( - z ) = + Ο€ ⁒ i + arcsech ⁑ z hyperbolic-inverse-secant 𝑧 πœ‹ 𝑖 hyperbolic-inverse-secant 𝑧 {\displaystyle{\displaystyle\operatorname{arcsech}\left(-z\right)=+\pi i+% \operatorname{arcsech}z}} arcsech(- z)= + Pi*I + arcsech(z) ArcSech[- z]= + Pi*I + ArcSech[z] Failure Failure Error Error
4.37.E15 arccoth ⁑ ( - z ) = - arccoth ⁑ z hyperbolic-inverse-cotangent 𝑧 hyperbolic-inverse-cotangent 𝑧 {\displaystyle{\displaystyle\operatorname{arccoth}\left(-z\right)=-% \operatorname{arccoth}z}} arccoth(- z)= - arccoth(z) ArcCoth[- z]= - ArcCoth[z] Failure Successful
Fail
0.-3.141592654*I <- {z = 1/2}
 -
4.37.E16 arcsinh ⁑ z = ln ⁑ ( ( z 2 + 1 ) 1 / 2 + z ) hyperbolic-inverse-sine 𝑧 superscript superscript 𝑧 2 1 1 2 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z% \right)}} arcsinh(z)= ln(((z)^(2)+ 1)^(1/ 2)+ z) ArcSinh[z]= Log[((z)^(2)+ 1)^(1/ 2)+ z] Failure Successful Error -
4.37.E17 arcsinh ⁑ ( i ⁒ y ) = 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 - 1 ) 1 / 2 + y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}% \pi i+\ln\left((y^{2}-1)^{1/2}+y\right)}} arcsinh(I*y)=(1)/(2)*Pi*I + ln(((y)^(2)- 1)^(1/ 2)+ y) ArcSinh[I*y]=Divide[1,2]*Pi*I + Log[((y)^(2)- 1)^(1/ 2)+ y] Failure Failure Error Error
4.37.E17 arcsinh ⁑ ( i ⁒ y ) = 1 2 ⁒ Ο€ ⁒ i - ln ⁑ ( ( y 2 - 1 ) 1 / 2 + y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}% \pi i-\ln\left((y^{2}-1)^{1/2}+y\right)}} arcsinh(I*y)=(1)/(2)*Pi*I - ln(((y)^(2)- 1)^(1/ 2)+ y) ArcSinh[I*y]=Divide[1,2]*Pi*I - Log[((y)^(2)- 1)^(1/ 2)+ y] Failure Failure Error Error
4.37.E18 arcsinh ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 - 1 ) 1 / 2 - y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2% }\pi i+\ln\left((y^{2}-1)^{1/2}-y\right)}} arcsinh(I*y)= -(1)/(2)*Pi*I + ln(((y)^(2)- 1)^(1/ 2)- y) ArcSinh[I*y]= -Divide[1,2]*Pi*I + Log[((y)^(2)- 1)^(1/ 2)- y] Failure Failure Error Error
4.37.E18 arcsinh ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ ⁒ i - ln ⁑ ( ( y 2 - 1 ) 1 / 2 - y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2% }\pi i-\ln\left((y^{2}-1)^{1/2}-y\right)}} arcsinh(I*y)= -(1)/(2)*Pi*I - ln(((y)^(2)- 1)^(1/ 2)- y) ArcSinh[I*y]= -Divide[1,2]*Pi*I - Log[((y)^(2)- 1)^(1/ 2)- y] Failure Failure Error Error
4.37.E19 arccosh ⁑ z = ln ⁑ ( + ( z 2 - 1 ) 1 / 2 + z ) hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}z=\ln\left(+(z^{2}-1)^{1/2}+% z\right)}} arccosh(z)= ln(+((z)^(2)- 1)^(1/ 2)+ z) ArcCosh[z]= Log[+((z)^(2)- 1)^(1/ 2)+ z] Failure Failure Error Error
4.37.E19 arccosh ⁑ z = ln ⁑ ( - ( z 2 - 1 ) 1 / 2 + z ) hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}z=\ln\left(-(z^{2}-1)^{1/2}+% z\right)}} arccosh(z)= ln(-((z)^(2)- 1)^(1/ 2)+ z) ArcCosh[z]= Log[-((z)^(2)- 1)^(1/ 2)+ z] Failure Failure Error Error
4.37.E20 arccosh ⁑ ( i ⁒ y ) = + 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 + 1 ) 1 / 2 + y ) hyperbolic-inverse-cosine imaginary-unit 𝑦 1 2 πœ‹ imaginary-unit superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arccosh}\left(\mathrm{i}y\right)=+% \tfrac{1}{2}\pi\mathrm{i}+\ln\left((y^{2}+1)^{1/2}+y\right)}} arccosh(I*y)= +(1)/(2)*Pi*I + ln(((y)^(2)+ 1)^(1/ 2)+ y) ArcCosh[I*y]= +Divide[1,2]*Pi*I + Log[((y)^(2)+ 1)^(1/ 2)+ y] Failure Failure Successful Successful
4.37.E20 arccosh ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 + 1 ) 1 / 2 - y ) hyperbolic-inverse-cosine imaginary-unit 𝑦 1 2 πœ‹ imaginary-unit superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arccosh}\left(\mathrm{i}y\right)=-% \tfrac{1}{2}\pi\mathrm{i}+\ln\left((y^{2}+1)^{1/2}-y\right)}} arccosh(I*y)= -(1)/(2)*Pi*I + ln(((y)^(2)+ 1)^(1/ 2)- y) ArcCosh[I*y]= -Divide[1,2]*Pi*I + Log[((y)^(2)+ 1)^(1/ 2)- y] Failure Failure
Fail
.9624236498+3.141592654*I <- {y = 1/2}
Fail
Complex[0.9624236501192068, 3.141592653589793] <- {Rule[y, Rational[1, 2]]}
4.37.E21 arccosh ⁑ z = 2 ⁒ ln ⁑ ( ( z + 1 2 ) 1 / 2 + ( z - 1 2 ) 1 / 2 ) hyperbolic-inverse-cosine 𝑧 2 superscript 𝑧 1 2 1 2 superscript 𝑧 1 2 1 2 {\displaystyle{\displaystyle\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}% {2}\right)^{1/2}+\left(\frac{z-1}{2}\right)^{1/2}\right)}} arccosh(z)= 2*ln(((z + 1)/(2))^(1/ 2)+((z - 1)/(2))^(1/ 2)) ArcCosh[z]= 2*Log[(Divide[z + 1,2])^(1/ 2)+(Divide[z - 1,2])^(1/ 2)] Failure Failure Error Error
4.37.E22 arccosh ⁑ x = + ln ⁑ ( i ⁒ ( 1 - x 2 ) 1 / 2 + x ) hyperbolic-inverse-cosine π‘₯ 𝑖 superscript 1 superscript π‘₯ 2 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=+\ln\left(i(1-x^{2})^{1/2}% +x\right)}} arccosh(x)= + ln(I*(1 - (x)^(2))^(1/ 2)+ x) ArcCosh[x]= + Log[I*(1 - (x)^(2))^(1/ 2)+ x] Failure Failure Error Error
4.37.E22 arccosh ⁑ x = - ln ⁑ ( i ⁒ ( 1 - x 2 ) 1 / 2 + x ) hyperbolic-inverse-cosine π‘₯ 𝑖 superscript 1 superscript π‘₯ 2 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=-\ln\left(i(1-x^{2})^{1/2}% +x\right)}} arccosh(x)= - ln(I*(1 - (x)^(2))^(1/ 2)+ x) ArcCosh[x]= - Log[I*(1 - (x)^(2))^(1/ 2)+ x] Failure Failure Error Error
4.37.E23 arccosh ⁑ x = + Ο€ ⁒ i + ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) hyperbolic-inverse-cosine π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=+\pi i+\ln\left((x^{2}-1)^% {1/2}-x\right)}} arccosh(x)= + Pi*I + ln(((x)^(2)- 1)^(1/ 2)- x) ArcCosh[x]= + Pi*I + Log[((x)^(2)- 1)^(1/ 2)- x] Failure Failure Error Error
4.37.E23 arccosh ⁑ x = - Ο€ ⁒ i + ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) hyperbolic-inverse-cosine π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=-\pi i+\ln\left((x^{2}-1)^% {1/2}-x\right)}} arccosh(x)= - Pi*I + ln(((x)^(2)- 1)^(1/ 2)- x) ArcCosh[x]= - Pi*I + Log[((x)^(2)- 1)^(1/ 2)- x] Failure Failure Error Error
4.37.E24 arctanh ⁑ z = 1 2 ⁒ ln ⁑ ( 1 + z 1 - z ) hyperbolic-inverse-tangent 𝑧 1 2 1 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac% {1+z}{1-z}\right)}} arctanh(z)=(1)/(2)*ln((1 + z)/(1 - z)) ArcTanh[z]=Divide[1,2]*Log[Divide[1 + z,1 - z]] Failure Failure Error Error
4.37.E25 arctanh ⁑ x = + 1 2 ⁒ Ο€ ⁒ i + 1 2 ⁒ ln ⁑ ( x + 1 x - 1 ) hyperbolic-inverse-tangent π‘₯ 1 2 πœ‹ 𝑖 1 2 π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle\operatorname{arctanh}x=+\tfrac{1}{2}\pi i+\tfrac{% 1}{2}\ln\left(\frac{x+1}{x-1}\right)}} arctanh(x)= +(1)/(2)*Pi*I +(1)/(2)*ln((x + 1)/(x - 1)) ArcTanh[x]= +Divide[1,2]*Pi*I +Divide[1,2]*Log[Divide[x + 1,x - 1]] Failure Failure Error Error
4.37.E25 arctanh ⁑ x = - 1 2 ⁒ Ο€ ⁒ i + 1 2 ⁒ ln ⁑ ( x + 1 x - 1 ) hyperbolic-inverse-tangent π‘₯ 1 2 πœ‹ 𝑖 1 2 π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle\operatorname{arctanh}x=-\tfrac{1}{2}\pi i+\tfrac{% 1}{2}\ln\left(\frac{x+1}{x-1}\right)}} arctanh(x)= -(1)/(2)*Pi*I +(1)/(2)*ln((x + 1)/(x - 1)) ArcTanh[x]= -Divide[1,2]*Pi*I +Divide[1,2]*Log[Divide[x + 1,x - 1]] Failure Failure Error Error
4.37.E26 z = sinh ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\sinh w}} z = sinh(w) z = Sinh[w] Failure Failure
Fail
1.112452092-.737321978*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.112452092-3.565749102*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.715975032-3.565749102*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.715975032-.737321978*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.1124520925053343, -0.7373219789661911] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.1124520925053343, -3.565749103712381] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.715975032240856, -3.565749103712381] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.715975032240856, -0.7373219789661911] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.37.E27 z = cosh ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\cosh w}} z = cosh(w) z = Cosh[w] Failure Failure
Fail
1.074539570-.497179547*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.074539570-3.325606671*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.753887554-3.325606671*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.753887554-.497179547*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.0745395706783705, -0.4971795477911152] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.0745395706783705, -3.3256066725373055] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7538875540678198, -3.3256066725373055] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7538875540678198, -0.4971795477911152] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.37.E28 z = tanh ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\tanh w}} z = tanh(w) z = Tanh[w] Failure Failure
Fail
.295839425+1.373342253*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.295839425-1.455084871*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-2.532587699-1.455084871*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-2.532587699+1.373342253*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.2958394249722609, 1.3733422538097753] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2958394249722609, -1.455084870936415] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.5325876997739294, -1.455084870936415] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.5325876997739294, 1.3733422538097753] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.37.E29 w = Arcsinh ⁑ z 𝑀 multivalued-hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arcsinh}z}} Error w = Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, z}] Error Failure -
Fail
Complex[0.022188930362652126, 0.6905247538249891] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.022188930362652126, 2.1379023709212013] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.806238194383538, 2.1379023709212013] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.806238194383538, 0.6905247538249891] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.37.E29 Arcsinh ⁑ z = ( - 1 ) k ⁒ arcsinh ⁑ z + k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-sine 𝑧 superscript 1 π‘˜ hyperbolic-inverse-sine 𝑧 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arcsinh}z=(-1)^{k}\operatorname{% arcsinh}z+k\pi i}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, z}]=(- 1)^(k)* ArcSinh[z]+ k*Pi*I Error Failure -
Fail
Complex[2.784049264020886, -1.694215036493581] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -6.283185307179586] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.784049264020886, -7.9774003436731675] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.784049264020886, -4.588970270686005] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.37.E30 w = Arccosh ⁑ z 𝑀 multivalued-hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arccosh}z}} Error w = Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, z}] Error Failure - Error
4.37.E30 Arccosh ⁑ z = + arccosh ⁑ z + 2 ⁒ k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-cosine 𝑧 hyperbolic-inverse-cosine 𝑧 2 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arccosh}z=+\operatorname{arccosh}z+2% k\pi i}} Error Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, z}]= + ArcCosh[z]+ 2*k*Pi*I Error Failure - Error
4.37.E30 Arccosh ⁑ z = - arccosh ⁑ z + 2 ⁒ k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-cosine 𝑧 hyperbolic-inverse-cosine 𝑧 2 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arccosh}z=-\operatorname{arccosh}z+2% k\pi i}} Error Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, z}]= - ArcCosh[z]+ 2*k*Pi*I Error Failure - Error
4.37.E31 w = Arctanh ⁑ z 𝑀 multivalued-hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arctanh}z}} Error w = Integrate[Divide[1, 1-t^2], {t, 0, z}] Error Failure -
Fail
Complex[0.8649074180390404, 1.4142135623730951] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
Complex[0.8649074180390404, -1.4142135623730951] <- {Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
Complex[-1.96351970670715, -1.4142135623730951] <- {Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
Complex[-1.96351970670715, 1.4142135623730951] <- {Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Rational[1, 2]]}
4.37.E31 Arctanh ⁑ z = arctanh ⁑ z + k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-tangent 𝑧 hyperbolic-inverse-tangent 𝑧 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arctanh}z=\operatorname{arctanh}z+k% \pi i}} Error Integrate[Divide[1, 1-t^2], {t, 0, z}]= ArcTanh[z]+ k*Pi*I Error Failure -
Fail
Complex[0.0, -3.141592653589793] <- {Rule[k, 1], Rule[z, Rational[1, 2]]}
Complex[0.0, -6.283185307179586] <- {Rule[k, 2], Rule[z, Rational[1, 2]]}
Complex[0.0, -9.42477796076938] <- {Rule[k, 3], Rule[z, Rational[1, 2]]}
4.38.E9 d d z ⁑ arcsinh ⁑ z = ( 1 + z 2 ) - 1 / 2 derivative 𝑧 hyperbolic-inverse-sine 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arcsinh}z=(1+z^{2})^{-1/2}}} diff(arcsinh(z), z)=(1 + (z)^(2))^(- 1/ 2) D[ArcSinh[z], z]=(1 + (z)^(2))^(- 1/ 2) Successful Successful - -
4.38.E10 d d z ⁑ arccosh ⁑ z = + ( z 2 - 1 ) - 1 / 2 derivative 𝑧 hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccosh}z=+(z^{2}-1)^{-1/2}}} diff(arccosh(z), z)= +((z)^(2)- 1)^(- 1/ 2) D[ArcCosh[z], z]= +((z)^(2)- 1)^(- 1/ 2) Failure Failure Successful Successful
4.38.E10 d d z ⁑ arccosh ⁑ z = - ( z 2 - 1 ) - 1 / 2 derivative 𝑧 hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccosh}z=-(z^{2}-1)^{-1/2}}} diff(arccosh(z), z)= -((z)^(2)- 1)^(- 1/ 2) D[ArcCosh[z], z]= -((z)^(2)- 1)^(- 1/ 2) Failure Failure
Fail
-2.309401076*I <- {z = 1/2}
Fail
Complex[0.0, -2.3094010767585034] <- {Rule[z, Rational[1, 2]]}
4.38.E11 d d z ⁑ arctanh ⁑ z = 1 1 - z 2 derivative 𝑧 hyperbolic-inverse-tangent 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arctanh}z=\frac{1}{1-z^{2}}}} diff(arctanh(z), z)=(1)/(1 - (z)^(2)) D[ArcTanh[z], z]=Divide[1,1 - (z)^(2)] Successful Successful - -
4.38.E12 d d z ⁑ arccsch ⁑ z = - 1 z ⁒ ( 1 + z 2 ) 1 / 2 derivative 𝑧 hyperbolic-inverse-cosecant 𝑧 1 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccsch}z=-\frac{1}{z(1+z^{2})^{1/2}}}} diff(arccsch(z), z)= -(1)/(z*(1 + (z)^(2))^(1/ 2)) D[ArcCsch[z], z]= -Divide[1,z*(1 + (z)^(2))^(1/ 2)] Failure Failure Successful Successful
4.38.E12 d d z ⁑ arccsch ⁑ z = + 1 z ⁒ ( 1 + z 2 ) 1 / 2 derivative 𝑧 hyperbolic-inverse-cosecant 𝑧 1 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccsch}z=+\frac{1}{z(1+z^{2})^{1/2}}}} diff(arccsch(z), z)= +(1)/(z*(1 + (z)^(2))^(1/ 2)) D[ArcCsch[z], z]= +Divide[1,z*(1 + (z)^(2))^(1/ 2)] Failure Failure
Fail
-3.577708764 <- {z = 1/2}
Fail
-3.5777087639996634 <- {Rule[z, Rational[1, 2]]}
4.38.E13 d d z ⁑ arcsech ⁑ z = - 1 z ⁒ ( 1 - z 2 ) 1 / 2 derivative 𝑧 hyperbolic-inverse-secant 𝑧 1 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arcsech}z=-\frac{1}{z(1-z^{2})^{1/2}}}} diff(arcsech(z), z)= -(1)/(z*(1 - (z)^(2))^(1/ 2)) D[ArcSech[z], z]= -Divide[1,z*(1 - (z)^(2))^(1/ 2)] Failure Failure Successful Successful
4.38.E14 d d z ⁑ arccoth ⁑ z = 1 1 - z 2 derivative 𝑧 hyperbolic-inverse-cotangent 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccoth}z=\frac{1}{1-z^{2}}}} diff(arccoth(z), z)=(1)/(1 - (z)^(2)) D[ArcCoth[z], z]=Divide[1,1 - (z)^(2)] Successful Successful - -
4.38.E15 Arcsinh ⁑ u + Arcsinh ⁑ v = Arcsinh ⁑ ( u ⁒ ( 1 + v 2 ) 1 / 2 + v ⁒ ( 1 + u 2 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑒 multivalued-hyperbolic-inverse-sine 𝑣 multivalued-hyperbolic-inverse-sine 𝑒 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑒 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arcsinh}v=% \operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}+v(1+u^{2})^{1/2}\right)}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u}]+ Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, v}]= Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u*(1 + (v)^(2))^(1/ 2)+ v*(1 + (u)^(2))^(1/ 2)}] Error Failure - Successful
4.38.E15 Arcsinh ⁑ u - Arcsinh ⁑ v = Arcsinh ⁑ ( u ⁒ ( 1 + v 2 ) 1 / 2 - v ⁒ ( 1 + u 2 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑒 multivalued-hyperbolic-inverse-sine 𝑣 multivalued-hyperbolic-inverse-sine 𝑒 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑒 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arcsinh}v=% \operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}-v(1+u^{2})^{1/2}\right)}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u}]- Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, v}]= Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u*(1 + (v)^(2))^(1/ 2)- v*(1 + (u)^(2))^(1/ 2)}] Error Failure - Skip
4.38.E16 Arccosh ⁑ u + Arccosh ⁑ v = Arccosh ⁑ ( u ⁒ v + ( ( u 2 - 1 ) ⁒ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-cosine 𝑒 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-cosine 𝑒 𝑣 superscript superscript 𝑒 2 1 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arccosh}u+\operatorname{Arccosh}v=% \operatorname{Arccosh}\left(uv+((u^{2}-1)(v^{2}-1))^{1/2}\right)}} Error Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, u}]+ Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, v}]= Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, u*v +(((u)^(2)- 1)*((v)^(2)- 1))^(1/ 2)}] Error Failure - Error
4.38.E16 Arccosh ⁑ u - Arccosh ⁑ v = Arccosh ⁑ ( u ⁒ v - ( ( u 2 - 1 ) ⁒ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-cosine 𝑒 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-cosine 𝑒 𝑣 superscript superscript 𝑒 2 1 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arccosh}u-\operatorname{Arccosh}v=% \operatorname{Arccosh}\left(uv-((u^{2}-1)(v^{2}-1))^{1/2}\right)}} Error Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, u}]- Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, v}]= Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, u*v -(((u)^(2)- 1)*((v)^(2)- 1))^(1/ 2)}] Error Failure - Error
4.38.E17 Arctanh ⁑ u + Arctanh ⁑ v = Arctanh ⁑ ( u + v 1 + u ⁒ v ) multivalued-hyperbolic-inverse-tangent 𝑒 multivalued-hyperbolic-inverse-tangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arctanh}v=% \operatorname{Arctanh}\left(\frac{u+v}{1+uv}\right)}} Error Integrate[Divide[1, 1-t^2], {t, 0, u}]+ Integrate[Divide[1, 1-t^2], {t, 0, v}]= Integrate[Divide[1, 1-t^2], {t, 0, Divide[u + v,1 + u*v]}] Error Failure - Skip
4.38.E17 Arctanh ⁑ u - Arctanh ⁑ v = Arctanh ⁑ ( u - v 1 - u ⁒ v ) multivalued-hyperbolic-inverse-tangent 𝑒 multivalued-hyperbolic-inverse-tangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑒 𝑣 1 𝑒 𝑣 {\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arctanh}v=% \operatorname{Arctanh}\left(\frac{u-v}{1-uv}\right)}} Error Integrate[Divide[1, 1-t^2], {t, 0, u}]- Integrate[Divide[1, 1-t^2], {t, 0, v}]= Integrate[Divide[1, 1-t^2], {t, 0, Divide[u - v,1 - u*v]}] Error Failure -
Fail
Complex[0.0, 1.6296538327419778] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.1102230246251565*^-16, 3.141592653589793] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -3.141592653589793] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.1102230246251565*^-16, -3.141592653589793] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.38.E18 Arcsinh ⁑ u + Arccosh ⁑ v = Arcsinh ⁑ ( u ⁒ v + ( ( 1 + u 2 ) ⁒ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑒 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arccosh}v=% \operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1))^{1/2}\right)}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u}]+ Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, v}]= Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/ 2)}] Error Failure - Error
4.38.E18 Arcsinh ⁑ u - Arccosh ⁑ v = Arcsinh ⁑ ( u ⁒ v - ( ( 1 + u 2 ) ⁒ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑒 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arccosh}v=% \operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1))^{1/2}\right)}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u}]- Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, v}]= Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/ 2)}] Error Failure - Error
4.38.E18 Arcsinh ⁑ ( u ⁒ v + ( ( 1 + u 2 ) ⁒ ( v 2 - 1 ) ) 1 / 2 ) = Arccosh ⁑ ( v ⁒ ( 1 + u 2 ) 1 / 2 + u ⁒ ( v 2 - 1 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 superscript 𝑣 2 1 1 2 multivalued-hyperbolic-inverse-cosine 𝑣 superscript 1 superscript 𝑒 2 1 2 𝑒 superscript superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1)% )^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}+u(v^{2}-1)^{1/2}% \right)}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/ 2)}]= Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, v*(1 + (u)^(2))^(1/ 2)+ u*((v)^(2)- 1)^(1/ 2)}] Error Failure - Error
4.38.E18 Arcsinh ⁑ ( u ⁒ v - ( ( 1 + u 2 ) ⁒ ( v 2 - 1 ) ) 1 / 2 ) = Arccosh ⁑ ( v ⁒ ( 1 + u 2 ) 1 / 2 - u ⁒ ( v 2 - 1 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑒 𝑣 superscript 1 superscript 𝑒 2 superscript 𝑣 2 1 1 2 multivalued-hyperbolic-inverse-cosine 𝑣 superscript 1 superscript 𝑒 2 1 2 𝑒 superscript superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1)% )^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}-u(v^{2}-1)^{1/2}% \right)}} Error Integrate[Divide[1, (1+t^2)^(1/2)], {t, 0, u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/ 2)}]= Integrate[Divide[1, (t^2-1)^(1/2)], {t, 1, v*(1 + (u)^(2))^(1/ 2)- u*((v)^(2)- 1)^(1/ 2)}] Error Failure - Error
4.38.E19 Arctanh ⁑ u + Arccoth ⁑ v = Arctanh ⁑ ( u ⁒ v + 1 v + u ) multivalued-hyperbolic-inverse-tangent 𝑒 multivalued-hyperbolic-inverse-cotangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 {\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arccoth}v=% \operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right)}} Error Integrate[Divide[1, 1-t^2], {t, 0, u}]+ Integrate[Divide[1, 1-t^2], {t, 0, Divide[1,v]}]= Integrate[Divide[1, 1-t^2], {t, 0, Divide[u*v + 1,v + u]}] Error Failure - Skip
4.38.E19 Arctanh ⁑ u - Arccoth ⁑ v = Arctanh ⁑ ( u ⁒ v - 1 v - u ) multivalued-hyperbolic-inverse-tangent 𝑒 multivalued-hyperbolic-inverse-cotangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 {\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arccoth}v=% \operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right)}} Error Integrate[Divide[1, 1-t^2], {t, 0, u}]- Integrate[Divide[1, 1-t^2], {t, 0, Divide[1,v]}]= Integrate[Divide[1, 1-t^2], {t, 0, Divide[u*v - 1,v - u]}] Error Failure -
Fail
Complex[5.551115123125783*^-17, -1.6296538327419778] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.282309879460564, -1.6296538327419778] <- {Rule[u, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.551115123125783*^-17, -1.6296538327419778] <- {Rule[u, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.282309879460564, 1.6296538327419778] <- {Rule[u, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
4.38.E19 Arctanh ⁑ ( u ⁒ v + 1 v + u ) = Arccoth ⁑ ( v + u u ⁒ v + 1 ) multivalued-hyperbolic-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 multivalued-hyperbolic-inverse-cotangent 𝑣 𝑒 𝑒 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right% )=\operatorname{Arccoth}\left(\frac{v+u}{uv+1}\right)}} Error Integrate[Divide[1, 1-t^2], {t, 0, Divide[u*v + 1,v + u]}]= Integrate[Divide[1, 1-t^2], {t, 0, Divide[1,Divide[v + u,u*v + 1]]}] Error Failure - Successful
4.38.E19 Arctanh ⁑ ( u ⁒ v - 1 v - u ) = Arccoth ⁑ ( v - u u ⁒ v - 1 ) multivalued-hyperbolic-inverse-tangent 𝑒 𝑣 1 𝑣 𝑒 multivalued-hyperbolic-inverse-cotangent 𝑣 𝑒 𝑒 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right% )=\operatorname{Arccoth}\left(\frac{v-u}{uv-1}\right)}} Error Integrate[Divide[1, 1-t^2], {t, 0, Divide[u*v - 1,v - u]}]= Integrate[Divide[1, 1-t^2], {t, 0, Divide[1,Divide[v - u,u*v - 1]]}] Error Failure - Error
4.40.E1 ∫ sinh ⁑ x ⁒ d x = cosh ⁑ x π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\sinh x\mathrm{d}x=\cosh x}} int(sinh(x), x)= cosh(x) Integrate[Sinh[x], x]= Cosh[x] Successful Successful - -
4.40.E2 ∫ cosh ⁑ x ⁒ d x = sinh ⁑ x π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\cosh x\mathrm{d}x=\sinh x}} int(cosh(x), x)= sinh(x) Integrate[Cosh[x], x]= Sinh[x] Successful Successful - -
4.40.E3 ∫ tanh ⁑ x ⁒ d x = ln ⁑ ( cosh ⁑ x ) π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\tanh x\mathrm{d}x=\ln\left(\cosh x\right)}} int(tanh(x), x)= ln(cosh(x)) Integrate[Tanh[x], x]= Log[Cosh[x]] Successful Successful - -
4.40.E4 ∫ csch ⁑ x ⁒ d x = ln ⁑ ( tanh ⁑ ( 1 2 ⁒ x ) ) π‘₯ π‘₯ 1 2 π‘₯ {\displaystyle{\displaystyle\int\operatorname{csch}x\mathrm{d}x=\ln\left(\tanh% \left(\tfrac{1}{2}x\right)\right)}} int(csch(x), x)= ln(tanh((1)/(2)*x)) Integrate[Csch[x], x]= Log[Tanh[Divide[1,2]*x]] Successful Successful - -
4.40.E5 ∫ sech ⁑ x ⁒ d x = gd ⁑ ( x ) π‘₯ π‘₯ Gudermannian π‘₯ {\displaystyle{\displaystyle\int\operatorname{sech}x\mathrm{d}x=\operatorname{% gd}\left(x\right)}} int(sech(x), x)= arctan(sinh(x)) Integrate[Sech[x], x]= Gudermannian[x] Successful Failure - Successful
4.40.E6 ∫ coth ⁑ x ⁒ d x = ln ⁑ ( sinh ⁑ x ) hyperbolic-cotangent π‘₯ π‘₯ π‘₯ {\displaystyle{\displaystyle\int\coth x\mathrm{d}x=\ln\left(\sinh x\right)}} int(coth(x), x)= ln(sinh(x)) Integrate[Coth[x], x]= Log[Sinh[x]] Successful Successful - -
4.40.E7 ∫ 0 ∞ e - x ⁒ sin ⁑ ( a ⁒ x ) sinh ⁑ x ⁒ d x = 1 2 ⁒ Ο€ ⁒ coth ⁑ ( 1 2 ⁒ Ο€ ⁒ a ) - 1 a superscript subscript 0 superscript 𝑒 π‘₯ π‘Ž π‘₯ π‘₯ π‘₯ 1 2 πœ‹ hyperbolic-cotangent 1 2 πœ‹ π‘Ž 1 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-x}\frac{\sin\left(ax\right)}{% \sinh x}\mathrm{d}x=\tfrac{1}{2}\pi\coth\left(\tfrac{1}{2}\pi a\right)-\frac{1% }{a}}} int(exp(- x)*(sin(a*x))/(sinh(x)), x = 0..infinity)=(1)/(2)*Pi*coth((1)/(2)*Pi*a)-(1)/(a) Integrate[Exp[- x]*Divide[Sin[a*x],Sinh[x]], {x, 0, Infinity}]=Divide[1,2]*Pi*Coth[Divide[1,2]*Pi*a]-Divide[1,a] Failure Failure Skip Successful
4.40.E8 ∫ 0 ∞ sinh ⁑ ( a ⁒ x ) sinh ⁑ ( Ο€ ⁒ x ) ⁒ d x = 1 2 ⁒ tan ⁑ ( 1 2 ⁒ a ) superscript subscript 0 π‘Ž π‘₯ πœ‹ π‘₯ π‘₯ 1 2 1 2 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\sinh\left(ax\right)}{\sinh% \left(\pi x\right)}\mathrm{d}x=\tfrac{1}{2}\tan\left(\tfrac{1}{2}a\right)}} int((sinh(a*x))/(sinh(Pi*x)), x = 0..infinity)=(1)/(2)*tan((1)/(2)*a) Integrate[Divide[Sinh[a*x],Sinh[Pi*x]], {x, 0, Infinity}]=Divide[1,2]*Tan[Divide[1,2]*a] Failure Failure Skip Successful
4.40.E9 ∫ - ∞ ∞ e a ⁒ x ( cosh ⁑ ( 1 2 ⁒ x ) ) 2 ⁒ d x = 4 ⁒ Ο€ ⁒ a sin ⁑ ( Ο€ ⁒ a ) superscript subscript superscript 𝑒 π‘Ž π‘₯ superscript 1 2 π‘₯ 2 π‘₯ 4 πœ‹ π‘Ž πœ‹ π‘Ž {\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh% \left(\tfrac{1}{2}x\right)\right)^{2}}\mathrm{d}x=\frac{4\pi a}{\sin\left(\pi a% \right)}}} int((exp(a*x))/((cosh((1)/(2)*x))^(2)), x = - infinity..infinity)=(4*Pi*a)/(sin(Pi*a)) Integrate[Divide[Exp[a*x],(Cosh[Divide[1,2]*x])^(2)], {x, - Infinity, Infinity}]=Divide[4*Pi*a,Sin[Pi*a]] Failure Failure Skip Skip
4.40.E11 ∫ arcsinh ⁑ x ⁒ d x = x ⁒ arcsinh ⁑ x - ( 1 + x 2 ) 1 / 2 hyperbolic-inverse-sine π‘₯ π‘₯ π‘₯ hyperbolic-inverse-sine π‘₯ superscript 1 superscript π‘₯ 2 1 2 {\displaystyle{\displaystyle\int\operatorname{arcsinh}x\mathrm{d}x=x% \operatorname{arcsinh}x-(1+x^{2})^{1/2}}} int(arcsinh(x), x)= x*arcsinh(x)-(1 + (x)^(2))^(1/ 2) Integrate[ArcSinh[x], x]= x*ArcSinh[x]-(1 + (x)^(2))^(1/ 2) Successful Successful - -
4.40.E12 ∫ arccosh ⁑ x ⁒ d x = x ⁒ arccosh ⁑ x - ( x 2 - 1 ) 1 / 2 hyperbolic-inverse-cosine π‘₯ π‘₯ π‘₯ hyperbolic-inverse-cosine π‘₯ superscript superscript π‘₯ 2 1 1 2 {\displaystyle{\displaystyle\int\operatorname{arccosh}x\mathrm{d}x=x% \operatorname{arccosh}x-(x^{2}-1)^{1/2}}} int(arccosh(x), x)= x*arccosh(x)-((x)^(2)- 1)^(1/ 2) Integrate[ArcCosh[x], x]= x*ArcCosh[x]-((x)^(2)- 1)^(1/ 2) Failure Successful Skip -
4.40.E13 ∫ arctanh ⁑ x ⁒ d x = x ⁒ arctanh ⁑ x + 1 2 ⁒ ln ⁑ ( 1 - x 2 ) hyperbolic-inverse-tangent π‘₯ π‘₯ π‘₯ hyperbolic-inverse-tangent π‘₯ 1 2 1 superscript π‘₯ 2 {\displaystyle{\displaystyle\int\operatorname{arctanh}x\mathrm{d}x=x% \operatorname{arctanh}x+\tfrac{1}{2}\ln\left(1-x^{2}\right)}} int(arctanh(x), x)= x*arctanh(x)+(1)/(2)*ln(1 - (x)^(2)) Integrate[ArcTanh[x], x]= x*ArcTanh[x]+Divide[1,2]*Log[1 - (x)^(2)] Successful Successful - -
4.40.E14 ∫ arccsch ⁑ x ⁒ d x = x ⁒ arccsch ⁑ x + arcsinh ⁑ x hyperbolic-inverse-cosecant π‘₯ π‘₯ π‘₯ hyperbolic-inverse-cosecant π‘₯ hyperbolic-inverse-sine π‘₯ {\displaystyle{\displaystyle\int\operatorname{arccsch}x\mathrm{d}x=x% \operatorname{arccsch}x+\operatorname{arcsinh}x}} int(arccsch(x), x)= x*arccsch(x)+ arcsinh(x) Integrate[ArcCsch[x], x]= x*ArcCsch[x]+ ArcSinh[x] Failure Successful Skip -
4.40.E15 ∫ arcsech ⁑ x ⁒ d x = x ⁒ arcsech ⁑ x + arcsin ⁑ x hyperbolic-inverse-secant π‘₯ π‘₯ π‘₯ hyperbolic-inverse-secant π‘₯ π‘₯ {\displaystyle{\displaystyle\int\operatorname{arcsech}x\mathrm{d}x=x% \operatorname{arcsech}x+\operatorname{arcsin}x}} int(arcsech(x), x)= x*arcsech(x)+ arcsin(x) Integrate[ArcSech[x], x]= x*ArcSech[x]+ ArcSin[x] Failure Successful Skip -
4.40.E16 ∫ arccoth ⁑ x ⁒ d x = x ⁒ arccoth ⁑ x + 1 2 ⁒ ln ⁑ ( x 2 - 1 ) hyperbolic-inverse-cotangent π‘₯ π‘₯ π‘₯ hyperbolic-inverse-cotangent π‘₯ 1 2 superscript π‘₯ 2 1 {\displaystyle{\displaystyle\int\operatorname{arccoth}x\mathrm{d}x=x% \operatorname{arccoth}x+\tfrac{1}{2}\ln\left(x^{2}-1\right)}} int(arccoth(x), x)= x*arccoth(x)+(1)/(2)*ln((x)^(2)- 1) Integrate[ArcCoth[x], x]= x*ArcCoth[x]+Divide[1,2]*Log[(x)^(2)- 1] Successful Failure -
Fail
Complex[0.0, 1.5707963267948966] <- {Rule[x, 2]}
Complex[0.0, 1.5707963267948966] <- {Rule[x, 3]}
4.42.E1 sin ⁑ A = a c 𝐴 π‘Ž 𝑐 {\displaystyle{\displaystyle\sin A=\frac{a}{c}}} sin(A)=(a)/(c) Sin[A]=Divide[a,c] Failure Failure
Fail
1.151535540+.3017614705*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
2.151535540-.6982385295*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
3.151535540+.3017614705*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
2.151535540+1.301761470*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.1515355413392863, 0.30176146986776087] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.1515355413392863, -0.6982385301322391] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.1515355413392863, 0.30176146986776087] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.1515355413392863, 1.3017614698677609] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E1 a c = 1 csc ⁑ A π‘Ž 𝑐 1 𝐴 {\displaystyle{\displaystyle\frac{a}{c}=\frac{1}{\csc A}}} (a)/(c)=(1)/(csc(A)) Divide[a,c]=Divide[1,Csc[A]] Failure Failure
Fail
-1.151535541-.3017614705*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
-2.151535541+.6982385295*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
-3.151535541-.3017614705*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
-2.151535541-1.301761470*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-1.1515355413392863, -0.30176146986776087] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.1515355413392863, 0.6982385301322391] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.1515355413392863, -0.30176146986776087] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.1515355413392863, -1.3017614698677609] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E2 cos ⁑ A = b c 𝐴 𝑏 𝑐 {\displaystyle{\displaystyle\cos A=\frac{b}{c}}} cos(A)=(b)/(c) Cos[A]=Divide[b,c] Failure Failure
Fail
-.6603260076-1.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
.3396739924-2.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
1.339673992-1.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
.3396739924-.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.6603260083052754, -1.9113931101642103] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.33967399169472456, -2.9113931101642105] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.3396739916947245, -1.9113931101642103] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.33967399169472456, -0.9113931101642103] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E2 b c = 1 sec ⁑ A 𝑏 𝑐 1 𝐴 {\displaystyle{\displaystyle\frac{b}{c}=\frac{1}{\sec A}}} (b)/(c)=(1)/(sec(A)) Divide[b,c]=Divide[1,Sec[A]] Failure Failure
Fail
.6603260076+1.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
-.3396739924+2.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
-1.339673992+1.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
-.3396739924+.911393109*I <- {A = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.6603260083052754, 1.9113931101642103] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.33967399169472456, 2.9113931101642105] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.3396739916947245, 1.9113931101642103] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.33967399169472456, 0.9113931101642103] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E3 tan ⁑ A = a b 𝐴 π‘Ž 𝑏 {\displaystyle{\displaystyle\tan A=\frac{a}{b}}} tan(A)=(a)/(b) Tan[A]=Divide[a,b] Failure Failure
Fail
-.9591286913+1.118374137*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2)}
.4087130869e-1+.118374137*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2)}
1.040871309+1.118374137*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2)}
.4087130869e-1+2.118374137*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.9591286914366802, 1.1183741374008342] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.04087130856331978, 0.11837413740083425] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.0408713085633199, 1.1183741374008342] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.04087130856331978, 2.118374137400834] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E3 a b = 1 cot ⁑ A π‘Ž 𝑏 1 𝐴 {\displaystyle{\displaystyle\frac{a}{b}=\frac{1}{\cot A}}} (a)/(b)=(1)/(cot(A)) Divide[a,b]=Divide[1,Cot[A]] Failure Failure
Fail
.9591286913-1.118374138*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2)}
-.4087130870e-1-.118374138*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2)}
-1.040871309-1.118374138*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2)}
-.4087130870e-1-2.118374138*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.9591286914366802, -1.1183741374008342] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.04087130856331978, -0.11837413740083425] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.0408713085633199, -1.1183741374008342] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.04087130856331978, -2.118374137400834] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E4 a sin ⁑ A = b sin ⁑ B π‘Ž 𝐴 𝑏 𝐡 {\displaystyle{\displaystyle\frac{a}{\sin A}=\frac{b}{\sin B}}} (a)/(sin(A))=(b)/(sin(B)) Divide[a,Sin[A]]=Divide[b,Sin[B]] Failure Failure
Fail
.1808221319+1.289247572*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2)}
1.470069704+1.108425440*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2)}
1.289247572-.1808221319*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)+I*2^(1/2)}
-.1808221319-1.289247572*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)-I*2^(1/2), b = 2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.18082213138216385, -0.18082213138216385] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4700697033135184, 1.1084254405491907] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.2892475719313545, 1.2892475719313545] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.18082213138216377, 1.2892475719313545] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E4 b sin ⁑ B = c sin ⁑ C 𝑏 𝐡 𝑐 𝐢 {\displaystyle{\displaystyle\frac{b}{\sin B}=\frac{c}{\sin C}}} (b)/(sin(B))=(c)/(sin(C)) Divide[b,Sin[B]]=Divide[c,Sin[C]] Failure Failure
Fail
.1808221319+1.289247572*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
1.470069704+1.108425440*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
1.289247572-.1808221319*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
-.1808221319-1.289247572*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.18082213138216385, -0.18082213138216385] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4700697033135184, 1.1084254405491907] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.2892475719313545, 1.2892475719313545] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.18082213138216377, 1.2892475719313545] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E5 c 2 = a 2 + b 2 - 2 ⁒ a ⁒ b ⁒ cos ⁑ C superscript 𝑐 2 superscript π‘Ž 2 superscript 𝑏 2 2 π‘Ž 𝑏 𝐢 {\displaystyle{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C}} (c)^(2)= (a)^(2)+ (b)^(2)- 2*a*b*cos(C) (c)^(2)= (a)^(2)+ (b)^(2)- 2*a*b*Cos[C] Failure Failure
Fail
15.29114486-1.282608060*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
15.29114486-9.282608052*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
15.29114486-1.282608060*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
15.29114486-9.282608052*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[15.291144881313683, -1.2826080664422035] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-15.291144881313683, -1.2826080664422035] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[15.291144881313683, -1.2826080664422035] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-15.291144881313683, -1.2826080664422035] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E6 a = b ⁒ cos ⁑ C + c ⁒ cos ⁑ B π‘Ž 𝑏 𝐢 𝑐 𝐡 {\displaystyle{\displaystyle a=b\cos C+c\cos B}} a = b*cos(C)+ c*cos(B) a = b*Cos[C]+ c*Cos[B] Failure Failure Skip Skip
4.42.E7 area = 1 2 ⁒ b ⁒ c ⁒ sin ⁑ A area 1 2 𝑏 𝑐 𝐴 {\displaystyle{\displaystyle\hbox{area}=\tfrac{1}{2}bc\sin A}} a*r*exp(1)*a*=(1)/(2)*b*c*sin(A) a*r*E*a*=Divide[1,2]*b*c*Sin[A] Failure Failure Skip Skip
4.42.E7 1 2 ⁒ b ⁒ c ⁒ sin ⁑ A = ( s ⁒ ( s - a ) ⁒ ( s - b ) ⁒ ( s - c ) ) 1 / 2 1 2 𝑏 𝑐 𝐴 superscript 𝑠 𝑠 π‘Ž 𝑠 𝑏 𝑠 𝑐 1 2 {\displaystyle{\displaystyle\tfrac{1}{2}bc\sin A=\left(s(s-a)(s-b)(s-c)\right)% ^{1/2}}} (1)/(2)*b*c*sin(A)=(s*(s - a)*(s - b)*(s - c))^(1/ 2) Divide[1,2]*b*c*Sin[A]=(s*(s - a)*(s - b)*(s - c))^(1/ 2) Failure Failure Skip Skip
4.42.E8 cos ⁑ a = cos ⁑ b ⁒ cos ⁑ c + sin ⁑ b ⁒ sin ⁑ c ⁒ cos ⁑ A π‘Ž 𝑏 𝑐 𝑏 𝑐 𝐴 {\displaystyle{\displaystyle\cos a=\cos b\cos c+\sin b\sin c\cos A}} cos(a)= cos(b)*cos(c)+ sin(b)*sin(c)*cos(A) Cos[a]= Cos[b]*Cos[c]+ Sin[b]*Sin[c]*Cos[A] Failure Failure
Fail
-.145682713+7.620029224*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
-5.032445392+7.110698060*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
7.901121089-8.845813328*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
-1.825810700-10.93348428*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.14568270504338976, 7.6200292388764925] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.032445395391095, 7.11069807526626] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[7.901121090331609, -8.845813349495826] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.8258107056535384, -10.933484295594681] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E9 sin ⁑ A sin ⁑ a = sin ⁑ B sin ⁑ b 𝐴 π‘Ž 𝐡 𝑏 {\displaystyle{\displaystyle\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}}} (sin(A))/(sin(a))=(sin(B))/(sin(b)) Divide[Sin[A],Sin[a]]=Divide[Sin[B],Sin[b]] Failure Failure
Fail
.385833893e-1-.2750965298*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2)}
2. <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2)}
1.961416611+.2750965298*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)+I*2^(1/2)}
-.385833893e-1+.2750965298*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), a = 2^(1/2)-I*2^(1/2), b = 2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.03858338910209658, 0.2750965290395158] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
2.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.9614166108979034, -0.2750965290395158] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.03858338910209658, -0.2750965290395158] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E9 sin ⁑ B sin ⁑ b = sin ⁑ C sin ⁑ c 𝐡 𝑏 𝐢 𝑐 {\displaystyle{\displaystyle\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}}} (sin(B))/(sin(b))=(sin(C))/(sin(c)) Divide[Sin[B],Sin[b]]=Divide[Sin[C],Sin[c]] Failure Failure
Fail
.385833893e-1-.2750965298*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = 2^(1/2)-I*2^(1/2)}
2. <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)-I*2^(1/2)}
1.961416611+.2750965298*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), c = -2^(1/2)+I*2^(1/2)}
-.385833893e-1+.2750965298*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), c = 2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.03858338910209658, 0.2750965290395158] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
2.0 <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.9614166108979034, -0.2750965290395158] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.03858338910209658, -0.2750965290395158] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E10 sin ⁑ a ⁒ cos ⁑ B = cos ⁑ b ⁒ sin ⁑ c - sin ⁑ b ⁒ cos ⁑ c ⁒ cos ⁑ A π‘Ž 𝐡 𝑏 𝑐 𝑏 𝑐 𝐴 {\displaystyle{\displaystyle\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A}} sin(a)*cos(B)= cos(b)*sin(c)- sin(b)*cos(c)*cos(A) Sin[a]*Cos[B]= Cos[b]*Sin[c]- Sin[b]*Cos[c]*Cos[A] Failure Failure Skip Skip
4.42.E11 cos ⁑ a ⁒ cos ⁑ C = sin ⁑ a ⁒ cot ⁑ b - sin ⁑ C ⁒ cot ⁑ B π‘Ž 𝐢 π‘Ž 𝑏 𝐢 𝐡 {\displaystyle{\displaystyle\cos a\cos C=\sin a\cot b-\sin C\cot B}} cos(a)*cos(C)= sin(a)*cot(b)- sin(C)*cot(B) Cos[a]*Cos[C]= Sin[a]*Cot[b]- Sin[C]*Cot[B] Failure Failure
Fail
-3.538045196-1.298501057*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2)}
-2.999121811-5.140982387*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2)}
-2.858697211-5.121287275*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2)}
-3.397620597-1.278805945*I <- {B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-3.538045200949385, -1.2985010548545435] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.229878658191402, -0.019695112023684347] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.217393184338834, 2.524285165473877] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.628377442853231, 3.842481332352105] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.42.E12 cos ⁑ A = - cos ⁑ B ⁒ cos ⁑ C + sin ⁑ B ⁒ sin ⁑ C ⁒ cos ⁑ a 𝐴 𝐡 𝐢 𝐡 𝐢 π‘Ž {\displaystyle{\displaystyle\cos A=-\cos B\cos C+\sin B\sin C\cos a}} cos(A)= - cos(B)*cos(C)+ sin(B)*sin(C)*cos(a) Cos[A]= - Cos[B]*Cos[C]+ Sin[B]*Sin[C]*Cos[a] Failure Failure
Fail
-7.221773105+5.023027110*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2)}
-2.257881161-12.32494952*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)-I*2^(1/2)}
-7.221773105+5.023027110*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = -2^(1/2)-I*2^(1/2)}
-2.257881161-12.32494952*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), C = 2^(1/2)+I*2^(1/2), a = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-7.22177310694216, 5.023027129167406] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.5051586890429873, 7.11069807526626] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.8250306884328387, -11.442815459204912] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[5.711793378780543, -10.933484295594681] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.45.E2 ln ⁑ x = 2 m ⁒ ln ⁑ ( 1 + y ) π‘₯ superscript 2 π‘š 1 𝑦 {\displaystyle{\displaystyle\ln x=2^{m}\ln\left(1+y\right)}} ln(x)= (2)^(m)* ln(1 + y) Log[x]= (2)^(m)* Log[1 + y] Failure Failure
Fail
-1.386294361 <- {m = 1, x = 1, y = 1}
-2.197224578 <- {m = 1, x = 1, y = 2}
-2.772588722 <- {m = 1, x = 1, y = 3}
-.6931471804 <- {m = 1, x = 2, y = 1}
... skip entries to safe data
Fail
-1.3862943611198906 <- {Rule[m, 1], Rule[x, 1], Rule[y, 1]}
-2.1972245773362196 <- {Rule[m, 1], Rule[x, 1], Rule[y, 2]}
-2.772588722239781 <- {Rule[m, 1], Rule[x, 1], Rule[y, 3]}
-0.6931471805599453 <- {Rule[m, 1], Rule[x, 2], Rule[y, 1]}
... skip entries to safe data
4.45.E3 ln ⁑ x = ln ⁑ ΞΎ + m ⁒ ln ⁑ 10 π‘₯ πœ‰ π‘š 10 {\displaystyle{\displaystyle\ln x=\ln\xi+m\ln 10}} ln(x)= ln(xi)+ m*ln(10) Log[x]= Log[\[Xi]]+ m*Log[10] Failure Failure
Fail
-2.995732273-.7853981634*I <- {xi = 2^(1/2)+I*2^(1/2), m = 1, x = 1}
-2.302585093-.7853981634*I <- {xi = 2^(1/2)+I*2^(1/2), m = 1, x = 2}
-1.897119984-.7853981634*I <- {xi = 2^(1/2)+I*2^(1/2), m = 1, x = 3}
-5.298317366-.7853981634*I <- {xi = 2^(1/2)+I*2^(1/2), m = 2, x = 1}
... skip entries to safe data
Fail
Complex[-2.9957322735539913, -0.7853981633974483] <- {Rule[m, 1], Rule[x, 1], Rule[ΞΎ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.302585092994046, -0.7853981633974483] <- {Rule[m, 1], Rule[x, 2], Rule[ΞΎ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.8971199848858813, -0.7853981633974483] <- {Rule[m, 1], Rule[x, 3], Rule[ΞΎ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.298317366548037, -0.7853981633974483] <- {Rule[m, 2], Rule[x, 1], Rule[ΞΎ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.45#Ex1 m = ⌊ x ln ⁑ 10 + 1 2 βŒ‹ π‘š π‘₯ 10 1 2 {\displaystyle{\displaystyle m=\left\lfloor\frac{x}{\ln 10}+\frac{1}{2}\right% \rfloor}} m = floor((x)/(ln(10))+(1)/(2)) m = Floor[Divide[x,Log[10]]+Divide[1,2]] Failure Failure
Fail
1. <- {m = 1, x = 1}
2. <- {m = 2, x = 1}
1. <- {m = 2, x = 2}
1. <- {m = 2, x = 3}
... skip entries to safe data
Fail
1.0 <- {Rule[m, 1], Rule[x, 1]}
2.0 <- {Rule[m, 2], Rule[x, 1]}
1.0 <- {Rule[m, 2], Rule[x, 2]}
1.0 <- {Rule[m, 2], Rule[x, 3]}
... skip entries to safe data
4.45#Ex2 y = x - m ⁒ ln ⁑ 10 𝑦 π‘₯ π‘š 10 {\displaystyle{\displaystyle y=x-m\ln 10}} y = x - m*ln(10) y = x - m*Log[10] Failure Failure
Fail
2.302585093 <- {m = 1, x = 1, y = 1}
3.302585093 <- {m = 1, x = 1, y = 2}
4.302585093 <- {m = 1, x = 1, y = 3}
1.302585093 <- {m = 1, x = 2, y = 1}
... skip entries to safe data
Fail
2.302585092994046 <- {Rule[m, 1], Rule[x, 1], Rule[y, 1]}
3.302585092994046 <- {Rule[m, 1], Rule[x, 1], Rule[y, 2]}
4.302585092994046 <- {Rule[m, 1], Rule[x, 1], Rule[y, 3]}
1.302585092994046 <- {Rule[m, 1], Rule[x, 2], Rule[y, 1]}
... skip entries to safe data
4.45#Ex3 m = ⌊ ΞΎ + 1 2 βŒ‹ π‘š πœ‰ 1 2 {\displaystyle{\displaystyle m=\left\lfloor\xi+\tfrac{1}{2}\right\rfloor}} m = floor(xi +(1)/(2)) m = Floor[\[Xi]+Divide[1,2]] Failure Failure
Fail
-1.-1.*I <- {xi = 2^(1/2)+I*2^(1/2), m = 1}
0.-1.*I <- {xi = 2^(1/2)+I*2^(1/2), m = 2}
1.-1.*I <- {xi = 2^(1/2)+I*2^(1/2), m = 3}
-1.+2.*I <- {xi = 2^(1/2)-I*2^(1/2), m = 1}
... skip entries to safe data
Fail
Complex[0.0, -1.0] <- {Rule[m, 1], Rule[ΞΎ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.0, -1.0] <- {Rule[m, 2], Rule[ΞΎ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.0, -1.0] <- {Rule[m, 3], Rule[ΞΎ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 2.0] <- {Rule[m, 1], Rule[ΞΎ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.45#Ex5 sin ⁑ x = ( - 1 ) m ⁒ sin ⁑ ΞΈ π‘₯ superscript 1 π‘š πœƒ {\displaystyle{\displaystyle\sin x=(-1)^{m}\sin\theta}} sin(x)=(- 1)^(m)* sin(theta) Sin[x]=(- 1)^(m)* Sin[\[Theta]] Failure Failure
Fail
2.993006525+.3017614705*I <- {theta = 2^(1/2)+I*2^(1/2), m = 1, x = 1}
3.060832967+.3017614705*I <- {theta = 2^(1/2)+I*2^(1/2), m = 1, x = 2}
2.292655548+.3017614705*I <- {theta = 2^(1/2)+I*2^(1/2), m = 1, x = 3}
-1.310064555-.3017614705*I <- {theta = 2^(1/2)+I*2^(1/2), m = 2, x = 1}
... skip entries to safe data
Fail
Complex[2.993006526147183, 0.30176146986776087] <- {Rule[m, 1], Rule[x, 1], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.060832968164968, 0.30176146986776087] <- {Rule[m, 1], Rule[x, 2], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.2926555493991536, 0.30176146986776087] <- {Rule[m, 1], Rule[x, 3], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.3100645565313898, -0.30176146986776087] <- {Rule[m, 2], Rule[x, 1], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.45#Ex6 cos ⁑ x = ( - 1 ) m ⁒ cos ⁑ ΞΈ π‘₯ superscript 1 π‘š πœƒ {\displaystyle{\displaystyle\cos x=(-1)^{m}\cos\theta}} cos(x)=(- 1)^(m)* cos(theta) Cos[x]=(- 1)^(m)* Cos[\[Theta]] Failure Failure
Fail
.8799762983-1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), m = 1, x = 1}
-.764728441e-1-1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), m = 1, x = 2}
-.6503185042-1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), m = 1, x = 3}
.2006283135+1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), m = 2, x = 1}
... skip entries to safe data
Fail
Complex[0.8799762975628643, -1.9113931101642103] <- {Rule[m, 1], Rule[x, 1], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.07647284485241784, -1.9113931101642103] <- {Rule[m, 1], Rule[x, 2], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.6503185049057209, -1.9113931101642103] <- {Rule[m, 1], Rule[x, 3], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2006283141734152, 1.9113931101642103] <- {Rule[m, 2], Rule[x, 1], Rule[ΞΈ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.45.E8 2 ⁒ arctan ⁑ x 1 + ( 1 + x 2 ) 1 / 2 = arctan ⁑ x 2 π‘₯ 1 superscript 1 superscript π‘₯ 2 1 2 π‘₯ {\displaystyle{\displaystyle 2\operatorname{arctan}\frac{x}{1+(1+x^{2})^{1/2}}% =\operatorname{arctan}x}} 2*arctan((x)/(1 +(1 + (x)^(2))^(1/ 2)))= arctan(x) 2*ArcTan[Divide[x,1 +(1 + (x)^(2))^(1/ 2)]]= ArcTan[x] Successful Failure - Successful
4.45.E10 arctan ⁑ x = 2 n ⁒ arctan ⁑ x n π‘₯ superscript 2 𝑛 subscript π‘₯ 𝑛 {\displaystyle{\displaystyle\operatorname{arctan}x=2^{n}\operatorname{arctan}x% _{n}}} arctan(x)= (2)^(n)* arctan(x[n]) ArcTan[x]= (2)^(n)* ArcTan[Subscript[x, n]] Failure Failure
Fail
-1.600225078-.6411549398*I <- {x[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
-1.278474524-.6411549398*I <- {x[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
-1.136577470-.6411549398*I <- {x[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
-3.985848320-1.282309880*I <- {x[n] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
 Successful
4.45.E13 arctan ⁑ x = 16 ⁒ arctan ⁑ x 4 π‘₯ 16 subscript π‘₯ 4 {\displaystyle{\displaystyle\operatorname{arctan}x=16\operatorname{arctan}x_{4% }}} arctan(x)= 16*arctan(x[4]) ArcTan[x]= 16*ArcTan[Subscript[x, 4]] Failure Failure
Fail
-18.29958778-5.129239518*I <- {x[4] = 2^(1/2)+I*2^(1/2), x = 1}
-17.97783722-5.129239518*I <- {x[4] = 2^(1/2)+I*2^(1/2), x = 2}
-17.83594017-5.129239518*I <- {x[4] = 2^(1/2)+I*2^(1/2), x = 3}
-18.29958778+5.129239518*I <- {x[4] = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Successful
4.45.E15 ln ⁑ z = ln ⁑ | z | + i ⁒ ph ⁑ z 𝑧 𝑧 𝑖 phase 𝑧 {\displaystyle{\displaystyle\ln z=\ln|z|+i\operatorname{ph}z}} ln(z)= ln(abs(z))+ I*argument(z) Log[z]= Log[Abs[z]]+ I*Arg[z] Failure Successful Skip -
4.45.E16 e z = e β„œ ⁑ z ⁒ ( cos ⁑ ( β„‘ ⁑ z ) + i ⁒ sin ⁑ ( β„‘ ⁑ z ) ) superscript 𝑒 𝑧 superscript 𝑒 𝑧 𝑧 𝑖 𝑧 {\displaystyle{\displaystyle e^{z}=e^{\Re z}(\cos\left(\Im z\right)+i\sin\left% (\Im z\right))}} exp(z)= exp(Re(z))*(cos(Im(z))+ I*sin(Im(z))) Exp[z]= Exp[Re[z]]*(Cos[Im[z]]+ I*Sin[Im[z]]) Failure Successful Successful -
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