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)) Log[a,z]=Divide[Log[z],Log[a]] Successful Successful - -
4.2.E9 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)) Log[a,b]=Divide[1,Log[b,a]] Successful Successful - -
4.2.E12 ln(exp(1))= 1 Log[E]= 1 Successful Successful - -
4.2.E13 int((1)/(t), t = 1..exp(1))= 1 Integrate[Divide[1,t], {t, 1, E}]= 1 Successful Successful - -
4.2.E14 log[exp(1)](z)= ln(z) Log[E,z]= Log[z] Successful Successful - -
4.2.E15 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](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) Log[z]=(Log[10])* Log[10,z] Successful Successful - -
4.2.E20 exp(z + 2*Pi*I)= exp(z) Exp[z + 2*Pi*I]= Exp[z] Successful Successful - -
4.2.E21 exp(- z)= 1/ exp(z) Exp[- z]= 1/ Exp[z] Successful Successful - -
4.2.E22 abs(exp(z))= exp(Re(z)) Abs[Exp[z]]= Exp[Re[z]] Successful Successful - -
4.2.E23 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)= 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)) (z)^(a)= Exp[a*Log[z]] Successful Failure - Successful
4.2.E28 (z)^(a)= exp(a*ln(z)) (z)^(a)= Exp[a*Log[z]] Successful Successful - -
4.2.E29 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 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 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 exp(z)= exp(z) Exp[z]= Exp[z] Successful Successful - -
4.2.E33 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 - Pi < = Im((1)/(a)*ln(w)) - Pi < = Im[Divide[1,a]*Log[w]] Failure Failure Successful Successful
4.2.E36 Im((1)/(a)*ln(w))< = Pi Im[Divide[1,a]*Log[w]]< = Pi Failure Failure Successful Successful
4.4.E1 ln(1)= 0 Log[1]= 0 Successful Successful - -
4.4.E2 ln(- 1 + I*0)= + Pi*I Log[- 1 + I*0]= + Pi*I Successful Successful - -
4.4.E2 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)*Pi*I Log[+ I]= +Divide[1,2]*Pi*I Successful Successful - -
4.4.E3 ln(- I)= -(1)/(2)*Pi*I Log[- I]= -Divide[1,2]*Pi*I Successful Successful - -
4.4.E5 exp(+ Pi*I)= - 1 Exp[+ Pi*I]= - 1 Successful Successful - -
4.4.E5 exp(- Pi*I)= - 1 Exp[- Pi*I]= - 1 Successful Successful - -
4.4.E6 exp(+ Pi*I/ 2)= + I Exp[+ Pi*I/ 2]= + I Successful Successful - -
4.4.E6 exp(- Pi*I/ 2)= - I Exp[- Pi*I/ 2]= - I Successful Successful - -
4.4.E7 exp(2*Pi*k*I)= 1 Exp[2*Pi*k*I]= 1 Successful Failure - Successful
4.4.E8 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 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 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 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 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 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 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 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)= exp(- Pi/ 2) (I)^(+ I)= Exp[- Pi/ 2] Successful Successful - -
4.4.E12 (I)^(- I)= exp(+ Pi/ 2) (I)^(- I)= Exp[+ Pi/ 2] Successful Successful - -
4.4.E13 limit((x)^(- a)* ln(x), x = infinity)= 0 Limit[(x)^(- a)* Log[x], x -> Infinity]= 0 Successful Failure - Successful
4.4.E14 limit((x)^(a)* ln(x), x = 0)= 0 Limit[(x)^(a)* Log[x], x -> 0]= 0 Failure Failure Skip Successful
4.4.E19 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) Divide[x,1 + x]< Log[1 + x] Failure Failure Skip Successful
4.5.E1 ln(1 + x)< x Log[1 + x]< x Failure Failure Skip Successful
4.5.E2 x < - ln(1 - x) x < - Log[1 - x] Failure Failure Skip Successful
4.5.E2 - ln(1 - x)<(x)/(1 - x) - Log[1 - x]<Divide[x,1 - x] Failure Failure Skip Successful
4.5.E3 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 Log[x]< = x - 1 Failure Failure Successful Successful
4.5.E5 ln(x)< = a*((x)^(1/ a)- 1) Log[x]< = a*((x)^(1/ a)- 1) Failure Failure Successful Successful
4.5.E6 abs(ln(1 + z))< = - ln(1 -abs(z)) Abs[Log[1 + z]]< = - Log[1 -Abs[z]] Failure Failure Successful Successful
4.7.E1 diff(ln(z), z)=(1)/(z) D[Log[z], z]=Divide[1,z] Successful Successful - -
4.7.E2 diff(ln(z), z)=(1)/(z) D[Log[z], z]=Divide[1,z] Successful Successful - -
4.7.E3 diff(ln(z), [z$(n)])=(- 1)^(n - 1)*factorial(n - 1)*(z)^(- n) D[Log[z], {z, n}]=(- 1)^(n - 1)*(n - 1)!*(z)^(- n) Failure Failure Successful Successful
4.7.E4 diff(ln(z), [z$(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 diff(exp(z), z)= exp(z) D[Exp[z], z]= Exp[z] Successful Successful - -
4.7.E8 diff(exp(a*z), z)= a*exp(a*z) D[Exp[a*z], z]= a*Exp[a*z] Successful Successful - -
4.7.E9 diff((a)^(z), z)= (a)^(z)* ln(a) D[(a)^(z), z]= (a)^(z)* Log[a] Successful Failure - Successful
4.7.E10 diff((z)^(a), z)= a*(z)^(a - 1) D[(z)^(a), z]= a*(z)^(a - 1) Successful Successful - -
4.7.E14 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]) 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]) 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]) 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]) 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) 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) Log[(z)^(n)]= n*Log[z] Failure Failure Skip Successful
4.8.E7 ln((1)/(z))= - ln(z) Log[Divide[1,z]]= - Log[z] Failure Failure Skip Successful
4.8.E8 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 Log[Exp[z]]= z Failure Failure Skip Successful
4.8.E10 exp(ln(z))= exp(ln(z)) Exp[Log[z]]= Exp[Log[z]] Successful Successful - -
4.8.E10 exp(ln(z))= z Exp[Log[z]]= z Successful Successful - -
4.8.E11 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*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) Log[(a)^(x)]= x*Log[a] Failure Failure Successful Successful
4.10.E1 int((1)/(z), z)= ln(z) Integrate[Divide[1,z], z]= Log[z] Successful Successful - -
4.10.E2 int(ln(z), z)= z*ln(z)- z Integrate[Log[z], z]= z*Log[z]- z Successful Successful - -
4.10.E3 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 int((1)/(z*ln(z)), z)= ln(ln(z)) Integrate[Divide[1,z*Log[z]], z]= Log[Log[z]] Successful Successful - -
4.10.E5 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 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 int(exp(a*z), z)=(exp(a*z))/(a) Integrate[Exp[a*z], z]=Divide[Exp[a*z],a] Successful Successful - -
4.10.E9 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 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 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 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 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 phi*(x)= ln(x + 1) \[Phi]*(x)= Log[x + 1] Failure Failure Skip Successful
4.12.E9 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[ℓ, 1], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.5657647880040868, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[ℓ, 2], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4712722490329502, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[ℓ, 3], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.45465367689297564, -2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[ℓ, 1], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
4.13.E1 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 LambertW(0, - 1/ exp(1))= LambertW(-1, - 1/ exp(1)) ProductLog[0, - 1/ E]= ProductLog[-1, - 1/ E] Successful Successful - -
4.13#Ex1 LambertW(-1, - 1/ exp(1))= - 1 ProductLog[-1, - 1/ E]= - 1 Successful Successful - -
4.13#Ex2 LambertW(0, 0)= 0 ProductLog[0, 0]= 0 Successful Successful - -
4.13#Ex3 LambertW(0, exp(1))= 1 ProductLog[0, E]= 1 Successful Successful - -
4.13#Ex4 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) 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)= 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 diff(LambertW(x), =)*(exp(- LambertW($0)))/(1 + LambertW($0)) D[ProductLog[x], =]*Divide[Exp[- ProductLog[$0]],1 + ProductLog[$0]] Error Error - -
4.13.E5 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 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)=(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)=(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)= exp(+ I*z) Cos[z]+ I*Sin[z]= Exp[+ I*z] Successful Successful - -
4.14.E3 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)) Tan[z]=Divide[Sin[z],Cos[z]] Successful Successful - -
4.14.E5 csc(z)=(1)/(sin(z)) Csc[z]=Divide[1,Sin[z]] Successful Successful - -
4.14.E6 sec(z)=(1)/(cos(z)) Sec[z]=Divide[1,Cos[z]] Successful Successful - -
4.14.E7 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)) Divide[Cos[z],Sin[z]]=Divide[1,Tan[z]] Successful Successful - -
4.14.E8 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*Pi)= cos(z) Cos[z + 2*k*Pi]= Cos[z] Failure Failure Successful Successful
4.14.E10 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)*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)*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)*Pi + I*y) Sec[x + I*y]= Csc[x +Divide[1,2]*Pi + I*y] Successful Successful - -
4.17.E1 limit((sin(z))/(z), z = 0)= 1 Limit[Divide[Sin[z],z], z -> 0]= 1 Successful Successful - -
4.17.E2 limit((tan(z))/(z), z = 0)= 1 Limit[Divide[Tan[z],z], z -> 0]= 1 Successful Successful - -
4.17.E3 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)/(Pi)< = sin(x) Divide[2*x,Pi]< = Sin[x] Failure Failure Skip Successful
4.18.E1 sin(x)< = x Sin[x]< = x Failure Failure Skip Successful
4.18.E2 x < = tan(x) x < = Tan[x] Failure Failure Skip Successful
4.18.E3 cos(x)< =(sin(x))/(x) Cos[x]< =Divide[Sin[x],x] Failure Failure Skip Successful
4.18.E3 (sin(x))/(x)< = 1 Divide[Sin[x],x]< = 1 Failure Failure Skip Successful
4.18.E4 Pi <(sin(Pi*x))/(x*(1 - x)) Pi <Divide[Sin[Pi*x],x*(1 - x)] Failure Failure Successful Successful
4.18.E4 (sin(Pi*x))/(x*(1 - x))< = 4 Divide[Sin[Pi*x],x*(1 - x)]< = 4 Failure Failure Successful Successful
4.18.E5 abs(sinh(y))< =abs(sin(z))<= cosh(y) Abs[Sinh[y]]< =Abs[Sin[z]]<= Cosh[y] Failure Failure Error Successful
4.18.E6 abs(sinh(y))< =abs(cos(z))<= cosh(y) Abs[Sinh[y]]< =Abs[Cos[z]]<= Cosh[y] Failure Failure Error Successful
4.18.E7 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 abs(cos(z))< = cosh(abs(z)) Abs[Cos[z]]< = Cosh[Abs[z]] Failure Failure Successful Successful
4.18.E9 abs(sin(z))< = sinh(abs(z)) Abs[Sin[z]]< = Sinh[Abs[z]] Failure Failure Successful Successful
4.18#Ex1 abs(cos(z))< 2 Abs[Cos[z]]< 2 Failure Failure Successful Successful
4.18#Ex2 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))= 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))= 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))= 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 diff(sin(z), z)= cos(z) D[Sin[z], z]= Cos[z] Successful Successful - -
4.20.E2 diff(cos(z), z)= - sin(z) D[Cos[z], z]= - Sin[z] Successful Successful - -
4.20.E3 diff(tan(z), z)= (sec(z))^(2) D[Tan[z], z]= (Sec[z])^(2) Successful Successful - -
4.20.E4 diff(csc(z), z)= - csc(z)*cot(z) D[Csc[z], z]= - Csc[z]*Cot[z] Successful Successful - -
4.20.E5 diff(sec(z), z)= sec(z)*tan(z) D[Sec[z], z]= Sec[z]*Tan[z] Successful Successful - -
4.20.E6 diff(cot(z), z)= - (csc(z))^(2) D[Cot[z], z]= - (Csc[z])^(2) Successful Successful - -
4.20.E7 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 diff(cos(z), [z$(n)])= cos(z +(1)/(2)*n*Pi) D[Cos[z], {z, n}]= Cos[z +Divide[1,2]*n*Pi] Successful Successful - -
4.20.E9 diff(w, [z$(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 (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 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) w = A*Cos[a*z]+ B*Sin[a*z] Failure Failure Skip Skip
4.20.E13 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) 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)=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)=sqrt(2)*sin(u -(1)/(4)*Pi) Sin[u]- Cos[u]=Sqrt[2]*Sin[u -Divide[1,4]*Pi] Successful Successful - -
4.21.E1 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 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) 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) 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) 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) 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)) 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)) 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)) 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)) 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)) 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)) 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)) 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)) 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)) 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)) 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)) 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)) Cot[u]- Cot[v]=Divide[Sin[v - u],Sin[u]*Sin[v]] Successful Successful - -
4.21.E12 (sin(z))^(2)+ (cos(z))^(2)= 1 (Sin[z])^(2)+ (Cos[z])^(2)= 1 Successful Successful - -
4.21.E13 (sec(z))^(2)= 1 + (tan(z))^(2) (Sec[z])^(2)= 1 + (Tan[z])^(2) Successful Successful - -
4.21.E14 (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*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*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*Sin[u]*Cos[v]= Sin[u - v]+ Sin[u + v] Successful Successful - -
4.21.E18 (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(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(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) 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) 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) 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) 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) 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) 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)) +(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)) -(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)) Divide[1 - Cos[z],Sin[z]]=Divide[Sin[z],1 + Cos[z]] Successful Successful - -
4.21.E24 sin(- z)= - sin(z) Sin[- z]= - Sin[z] Successful Successful - -
4.21.E25 cos(- z)= cos(z) Cos[- z]= Cos[z] Successful Successful - -
4.21.E26 tan(- z)= - tan(z) Tan[- z]= - Tan[z] Successful Successful - -
4.21.E27 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(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(z))^(2)- 1 Cos[2*z]= 2*(Cos[z])^(2)- 1 Successful Successful - -
4.21.E28