Results of Integrals with Coalescing Saddles

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
36.2#Ex2 F + ⁒ ( 𝐱 ) = ∫ 0 ∞ cos ⁑ ( r ⁒ y ⁒ exp ⁑ ( + i ⁒ Ο€ 6 ) ) ⁒ exp ⁑ ( 2 ⁒ i ⁒ r 2 ⁒ z ⁒ exp ⁑ ( + i ⁒ Ο€ 3 ) ) ⁒ Ai ⁑ ( 3 2 / 3 ⁒ r 2 + 3 - 1 / 3 ⁒ exp ⁑ ( - i ⁒ Ο€ 3 ) ⁒ ( 1 3 ⁒ z 2 - x ) ) ⁒ d r subscript F 𝐱 superscript subscript 0 π‘Ÿ 𝑦 𝑖 πœ‹ 6 2 𝑖 superscript π‘Ÿ 2 𝑧 𝑖 πœ‹ 3 Airy-Ai superscript 3 2 3 superscript π‘Ÿ 2 superscript 3 1 3 𝑖 πœ‹ 3 1 3 superscript 𝑧 2 π‘₯ π‘Ÿ {\displaystyle{\displaystyle\mathrm{F}_{+}(\mathbf{x})=\int_{0}^{\infty}\cos% \left(ry\exp\left(+i\dfrac{\pi}{6}\right)\right)\exp\left(2ir^{2}z\exp\left(+i% \dfrac{\pi}{3}\right)\right)\mathrm{Ai}\left(3^{2/3}r^{2}+3^{-1/3}\exp\left(-i% \dfrac{\pi}{3}\right)\left(\tfrac{1}{3}z^{2}-x\right)\right)\mathrm{d}r}} F[+]*(x)= int(cos(r*y*exp(+ I*(Pi)/(6)))*exp(2*I*(r)^(2)* z*exp(+ I*(Pi)/(3)))*AiryAi((3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* exp(- I*(Pi)/(3))*((1)/(3)*(z)^(2)- x)), r = 0..infinity) Subscript[F, +]*(x)= Integrate[Cos[r*y*Exp[+ I*Divide[Pi,6]]]*Exp[2*I*(r)^(2)* z*Exp[+ I*Divide[Pi,3]]]*AiryAi[(3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* Exp[- I*Divide[Pi,3]]*(Divide[1,3]*(z)^(2)- x)], {r, 0, Infinity}] Error Failure - Error
36.2#Ex2 F - ⁒ ( 𝐱 ) = ∫ 0 ∞ cos ⁑ ( r ⁒ y ⁒ exp ⁑ ( - i ⁒ Ο€ 6 ) ) ⁒ exp ⁑ ( 2 ⁒ i ⁒ r 2 ⁒ z ⁒ exp ⁑ ( - i ⁒ Ο€ 3 ) ) ⁒ Ai ⁑ ( 3 2 / 3 ⁒ r 2 + 3 - 1 / 3 ⁒ exp ⁑ ( + i ⁒ Ο€ 3 ) ⁒ ( 1 3 ⁒ z 2 - x ) ) ⁒ d r subscript F 𝐱 superscript subscript 0 π‘Ÿ 𝑦 𝑖 πœ‹ 6 2 𝑖 superscript π‘Ÿ 2 𝑧 𝑖 πœ‹ 3 Airy-Ai superscript 3 2 3 superscript π‘Ÿ 2 superscript 3 1 3 𝑖 πœ‹ 3 1 3 superscript 𝑧 2 π‘₯ π‘Ÿ {\displaystyle{\displaystyle\mathrm{F}_{-}(\mathbf{x})=\int_{0}^{\infty}\cos% \left(ry\exp\left(-i\dfrac{\pi}{6}\right)\right)\exp\left(2ir^{2}z\exp\left(-i% \dfrac{\pi}{3}\right)\right)\mathrm{Ai}\left(3^{2/3}r^{2}+3^{-1/3}\exp\left(+i% \dfrac{\pi}{3}\right)\left(\tfrac{1}{3}z^{2}-x\right)\right)\mathrm{d}r}} F[-]*(x)= int(cos(r*y*exp(- I*(Pi)/(6)))*exp(2*I*(r)^(2)* z*exp(- I*(Pi)/(3)))*AiryAi((3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* exp(+ I*(Pi)/(3))*((1)/(3)*(z)^(2)- x)), r = 0..infinity) Subscript[F, -]*(x)= Integrate[Cos[r*y*Exp[- I*Divide[Pi,6]]]*Exp[2*I*(r)^(2)* z*Exp[- I*Divide[Pi,3]]]*AiryAi[(3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* Exp[+ I*Divide[Pi,3]]*(Divide[1,3]*(z)^(2)- x)], {r, 0, Infinity}] Error Failure - Error
36.2.E14 P ⁒ ( x 2 , x 1 ) = ∫ - ∞ ∞ exp ⁑ ( i ⁒ ( t 4 + x 2 ⁒ t 2 + x 1 ⁒ t ) ) ⁒ d t 𝑃 subscript π‘₯ 2 subscript π‘₯ 1 superscript subscript imaginary-unit superscript 𝑑 4 subscript π‘₯ 2 superscript 𝑑 2 subscript π‘₯ 1 𝑑 𝑑 {\displaystyle{\displaystyle P(x_{2},x_{1})=\int_{-\infty}^{\infty}\exp\left(% \mathrm{i}(t^{4}+x_{2}t^{2}+x_{1}t)\right)\mathrm{d}t}} P*(x[2], x[1])= int(exp(I*((t)^(4)+ x[2]*(t)^(2)+ x[1]*t)), t = - infinity..infinity) P*(Subscript[x, 2], Subscript[x, 1])= Integrate[Exp[I*((t)^(4)+ Subscript[x, 2]*(t)^(2)+ Subscript[x, 1]*t)], {t, - Infinity, Infinity}] Failure Failure Skip Error
36.2#Ex10 1 3 ⁒ Ο€ ⁒ Ξ“ ⁑ ( 1 6 ) = 3.28868 1 3 πœ‹ Euler-Gamma 1 6 3.28868 {\displaystyle{\displaystyle\tfrac{1}{3}\sqrt{\pi}\Gamma\left(\tfrac{1}{6}% \right)=3.28868}} (1)/(3)*sqrt(Pi)*GAMMA((1)/(6))= 3.28868 Divide[1,3]*Sqrt[Pi]*Gamma[Divide[1,6]]= 3.28868 Successful Failure - Successful
36.2#Ex11 1 3 ⁒ Ξ“ 2 ⁑ ( 1 3 ) = 2.39224 1 3 Euler-Gamma 2 1 3 2.39224 {\displaystyle{\displaystyle\tfrac{1}{3}{\Gamma^{2}}\left(\tfrac{1}{3}\right)=% 2.39224}} (1)/(3)*(GAMMA((1)/(3)))^(2)= 2.39224 Divide[1,3]*(Gamma[Divide[1,3]])^(2)= 2.39224 Successful Failure - Successful
36.4#Ex11 x = 1 3 ⁒ z 2 ⁒ ( - cos ⁑ ( 2 ⁒ Ο• ) - 2 ⁒ cos ⁑ Ο• ) π‘₯ 1 3 superscript 𝑧 2 2 italic-Ο• 2 italic-Ο• {\displaystyle{\displaystyle x=\tfrac{1}{3}z^{2}(-\cos\left(2\phi\right)-2\cos% \phi)}} x =(1)/(3)*(z)^(2)*(- cos(2*phi)- 2*cos(phi)) x =Divide[1,3]*(z)^(2)*(- Cos[2*\[Phi]]- 2*Cos[\[Phi]]) Failure Failure
Fail
9.559717772-9.862323206*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 1}
10.55971777-9.862323206*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 2}
11.55971777-9.862323206*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 3}
-7.559717772+9.862323206*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[9.559717773383342, -9.862323224679095] <- {Rule[x, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[10.559717773383342, -9.862323224679095] <- {Rule[x, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[11.559717773383342, -9.862323224679095] <- {Rule[x, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-7.559717773383342, -9.862323224679095] <- {Rule[x, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
36.4#Ex12 y = 1 3 ⁒ z 2 ⁒ ( sin ⁑ ( 2 ⁒ Ο• ) - 2 ⁒ sin ⁑ Ο• ) 𝑦 1 3 superscript 𝑧 2 2 italic-Ο• 2 italic-Ο• {\displaystyle{\displaystyle y=\tfrac{1}{3}z^{2}(\sin\left(2\phi\right)-2\sin% \phi)}} y =(1)/(3)*(z)^(2)*(sin(2*phi)- 2*sin(phi)) y =Divide[1,3]*(z)^(2)*(Sin[2*\[Phi]]- 2*Sin[\[Phi]]) Failure Failure
Fail
-10.49784307+2.250480207*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 1}
-9.49784307+2.250480207*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 2}
-8.49784307+2.250480207*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 3}
12.49784307-2.250480207*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
Fail
Complex[-10.497843084926524, 2.2504802168566327] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-9.497843084926524, 2.2504802168566327] <- {Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-8.497843084926524, 2.2504802168566327] <- {Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[12.497843084926524, 2.2504802168566327] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο•, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
36.4.E11 x + i ⁒ y = - z 2 ⁒ exp ⁑ ( 2 3 ⁒ i ⁒ Ο€ ⁒ m ) π‘₯ 𝑖 𝑦 superscript 𝑧 2 2 3 𝑖 πœ‹ π‘š {\displaystyle{\displaystyle x+iy=-z^{2}\exp\left(\tfrac{2}{3}i\pi m\right)}} x + I*y = - (z)^(2)* exp((2)/(3)*I*Pi*m) x + I*y = - (z)^(2)* Exp[Divide[2,3]*I*Pi*m] Error Failure - Successful
36.4#Ex13 x = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( 2 ⁒ Ο„ ) + 2 ⁒ exp ⁑ ( - Ο„ ) ) π‘₯ 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle x=-\tfrac{1}{12}z^{2}(\exp\left(2\tau\right)+2% \exp\left(-\tau\right))}} x = -(1)/(12)*(z)^(2)*(exp(2*tau)+ 2*exp(- tau)) x = -Divide[1,12]*(z)^(2)*(Exp[2*\[Tau]]+ 2*Exp[- \[Tau]]) Failure Failure
Fail
-.577309391-5.340041571*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 1}
.422690609-5.340041571*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 2}
1.422690609-5.340041571*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 3}
2.577309391+5.340041571*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[-0.5773093891316772, -5.340041577401404] <- {Rule[x, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.4226906108683228, -5.340041577401404] <- {Rule[x, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4226906108683228, -5.340041577401404] <- {Rule[x, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.577309389131677, -5.340041577401404] <- {Rule[x, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
36.4#Ex13 x = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( 2 ⁒ Ο„ ) - 2 ⁒ exp ⁑ ( - Ο„ ) ) π‘₯ 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle x=-\tfrac{1}{12}z^{2}(\exp\left(2\tau\right)-2% \exp\left(-\tau\right))}} x = -(1)/(12)*(z)^(2)*(exp(2*tau)- 2*exp(- tau)) x = -Divide[1,12]*(z)^(2)*(Exp[2*\[Tau]]- 2*Exp[- \[Tau]]) Failure Failure
Fail
-.897499299-5.390591597*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 1}
.102500701-5.390591597*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 2}
1.102500701-5.390591597*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), x = 3}
2.897499299+5.390591597*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[-0.8974992973651121, -5.390591606504023] <- {Rule[x, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.10250070263488786, -5.390591606504023] <- {Rule[x, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.1025007026348879, -5.390591606504023] <- {Rule[x, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.897499297365112, -5.390591606504023] <- {Rule[x, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
36.4#Ex14 y = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( - 2 ⁒ Ο„ ) + 2 ⁒ exp ⁑ ( Ο„ ) ) 𝑦 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle y=-\tfrac{1}{12}z^{2}(\exp\left(-2\tau\right)+2% \exp\left(\tau\right))}} y = -(1)/(12)*(z)^(2)*(exp(- 2*tau)+ 2*exp(tau)) y = -Divide[1,12]*(z)^(2)*(Exp[- 2*\[Tau]]+ 2*Exp[\[Tau]]) Failure Failure
Fail
-1.702549495+.4088799658*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 1}
-.702549495+.4088799658*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 2}
.297450505+.4088799658*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 3}
3.702549495-.4088799658*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
Fail
Complex[-1.7025494975599944, 0.4088799650618581] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.7025494975599944, 0.4088799650618581] <- {Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2974505024400056, 0.4088799650618581] <- {Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.7025494975599944, 0.4088799650618581] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
36.4#Ex14 y = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( - 2 ⁒ Ο„ ) - 2 ⁒ exp ⁑ ( Ο„ ) ) 𝑦 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle y=-\tfrac{1}{12}z^{2}(\exp\left(-2\tau\right)-2% \exp\left(\tau\right))}} y = -(1)/(12)*(z)^(2)*(exp(- 2*tau)- 2*exp(tau)) y = -Divide[1,12]*(z)^(2)*(Exp[- 2*\[Tau]]- 2*Exp[\[Tau]]) Failure Failure
Fail
3.714688702-.4463673177*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 1}
4.714688702-.4463673177*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 2}
5.714688702-.4463673177*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), y = 3}
-1.714688702+.4463673177*I <- {tau = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
Fail
Complex[3.7146887044446677, -0.4463673170214556] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.714688704444668, -0.4463673170214556] <- {Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[5.714688704444668, -0.4463673170214556] <- {Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.7146887044446677, -0.4463673170214556] <- {Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο„, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
36.5.E7 X = 9 20 + 20 ⁒ u 4 - Y 2 20 ⁒ u 2 + 6 ⁒ u 2 ⁒ sign ⁑ ( z ) 𝑋 9 20 20 superscript 𝑒 4 superscript π‘Œ 2 20 superscript 𝑒 2 6 superscript 𝑒 2 sign 𝑧 {\displaystyle{\displaystyle X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^% {2}\operatorname{sign}\left(z\right)}} X =(9)/(20)+ 20*(u)^(4)-((Y)^(2))/(20*(u)^(2))+ 6*(u)^(2)* signum(z) X =Divide[9,20]+ 20*(u)^(4)-Divide[(Y)^(2),20*(u)^(2)]+ 6*(u)^(2)* Sign[z] Failure Failure
Fail
337.9847759-15.55634918*I <- {X = 2^(1/2)+I*2^(1/2), Y = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
304.0436505-15.55634918*I <- {X = 2^(1/2)+I*2^(1/2), Y = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
304.0436505+18.38477630*I <- {X = 2^(1/2)+I*2^(1/2), Y = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
337.9847759+18.38477630*I <- {X = 2^(1/2)+I*2^(1/2), Y = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[337.98477631085024, -15.556349186104047] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[X, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Y, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[304.04365081389597, -15.556349186104047] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[X, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Y, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[304.04365081389597, 18.38477631085024] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[X, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Y, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[337.98477631085024, 18.38477631085024] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[X, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Y, 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
36.5.E8 16 ⁒ u 5 - Y 2 10 ⁒ u + 4 ⁒ u 3 ⁒ sign ⁑ ( z ) - 3 10 ⁒ | Y | ⁒ sign ⁑ ( z ) + 4 ⁒ t 5 + 2 ⁒ t 3 ⁒ sign ⁑ ( z ) + | Y | ⁒ t 2 = 0 16 superscript 𝑒 5 superscript π‘Œ 2 10 𝑒 4 superscript 𝑒 3 sign 𝑧 3 10 π‘Œ sign 𝑧 4 superscript 𝑑 5 2 superscript 𝑑 3 sign 𝑧 π‘Œ superscript 𝑑 2 0 {\displaystyle{\displaystyle 16u^{5}-\frac{Y^{2}}{10u}+4u^{3}\operatorname{% sign}\left(z\right)-\frac{3}{10}|Y|\operatorname{sign}\left(z\right)+4t^{5}+2t% ^{3}\operatorname{sign}\left(z\right)+|Y|t^{2}=0}} 16*(u)^(5)-((Y)^(2))/(10*u)+ 4*(u)^(3)* signum(z)-(3)/(10)*abs(Y)* signum(z)+ 4*(t)^(5)+ 2*(t)^(3)* signum(z)+abs(Y)* (t)^(2)= 0 16*(u)^(5)-Divide[(Y)^(2),10*u]+ 4*(u)^(3)* Sign[z]-Divide[3,10]*Abs[Y]* Sign[z]+ 4*(t)^(5)+ 2*(t)^(3)* Sign[z]+Abs[Y]* (t)^(2)= 0 Failure Failure
Fail
-501.1140248-445.1140248*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-453.1140248-396.2654968*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-404.2654968-444.2654967*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-452.2654967-493.1140248*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
36.5.E9 t = - u + ( | Y | 10 ⁒ u - u 2 - 3 10 ⁒ sign ⁑ ( z ) ) 1 / 2 𝑑 𝑒 superscript π‘Œ 10 𝑒 superscript 𝑒 2 3 10 sign 𝑧 1 2 {\displaystyle{\displaystyle t=-u+\left(\dfrac{|Y|}{10u}-u^{2}-\dfrac{3}{10}% \operatorname{sign}\left(z\right)\right)^{1/2}}} t = - u +((abs(Y))/(10*u)- (u)^(2)-(3)/(10)*signum(z))^(1/ 2) t = - u +(Divide[Abs[Y],10*u]- (u)^(2)-Divide[3,10]*Sign[z])^(1/ 2) Failure Failure
Fail
1.389025232+4.316143214*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.464655141+4.243098598*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.387633249+4.167473156*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.315976323+4.244288951*I <- {Y = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
36.7.E3 3 ⁒ Ο€ ⁒ ( 8 ⁒ n + 5 ) 9 + 8 ⁒ ΞΎ n ⁒ ΞΎ n 3 / 2 = 27 16 ⁒ ( 3 2 ) 1 / 2 ⁒ ( ln ⁑ ( 1 ΞΎ n ) + 3 ⁒ ln ⁑ ( 3 2 ) ) 3 πœ‹ 8 𝑛 5 9 8 subscript πœ‰ 𝑛 superscript subscript πœ‰ 𝑛 3 2 27 16 superscript 3 2 1 2 1 subscript πœ‰ 𝑛 3 3 2 {\displaystyle{\displaystyle\frac{3\pi(8n+5)}{9+8\xi_{n}}\xi_{n}^{3/2}=\dfrac{% 27}{16}\left(\dfrac{3}{2}\right)^{1/2}\left(\ln\left(\frac{1}{\xi_{n}}\right)+% 3\ln\left(\dfrac{3}{2}\right)\right)}} (3*Pi*(8*n + 5))/(9 + 8*xi[n])*(xi[n])^(3/ 2)=(27)/(16)*((3)/(2))^(1/ 2)*(ln((1)/(xi[n]))+ 3*ln((3)/(2))) Divide[3*Pi*(8*n + 5),9 + 8*Subscript[\[Xi], n]]*(Subscript[\[Xi], n])^(3/ 2)=Divide[27,16]*(Divide[3,2])^(1/ 2)*(Log[Divide[1,Subscript[\[Xi], n]]]+ 3*Log[Divide[3,2]]) Failure Failure
Fail
10.60125548+10.87763041*I <- {xi[n] = 2^(1/2)+I*2^(1/2), n = 1}
17.79059838+16.57264781*I <- {xi[n] = 2^(1/2)+I*2^(1/2), n = 2}
24.97994128+22.26766522*I <- {xi[n] = 2^(1/2)+I*2^(1/2), n = 3}
10.60125548-10.87763041*I <- {xi[n] = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
 Successful
36.7.E6 exp ⁑ ( - 2 ⁒ Ο€ ⁒ i ⁒ ( z - z n Ξ” ⁒ z + 2 ⁒ x Ξ” ⁒ x ) ) ⁒ ( 2 ⁒ exp ⁑ ( - 6 ⁒ Ο€ ⁒ i ⁒ x Ξ” ⁒ x ) ⁒ cos ⁑ ( 2 ⁒ 3 ⁒ Ο€ ⁒ y Ξ” ⁒ x ) + 1 ) = 3 2 πœ‹ 𝑖 𝑧 subscript 𝑧 𝑛 Ξ” 𝑧 2 π‘₯ Ξ” π‘₯ 2 6 πœ‹ 𝑖 π‘₯ Ξ” π‘₯ 2 3 πœ‹ 𝑦 Ξ” π‘₯ 1 3 {\displaystyle{\displaystyle\exp\left(-2\pi i\left(\frac{z-z_{n}}{\Delta z}+% \frac{2x}{\Delta x}\right)\right)\*{\left(2\exp\left(\frac{-6\pi ix}{\Delta x}% \right)\cos\left(\frac{2\sqrt{3}\pi y}{\Delta x}\right)+1\right)}=\sqrt{3}}} exp(- 2*Pi*I*((z - z[n])/(Delta*z)+(2*x)/(Delta*x)))*(2*exp((- 6*Pi*I*x)/(Delta*x))*cos((2*sqrt(3)*Pi*y)/(Delta*x))+ 1)=sqrt(3) Exp[- 2*Pi*I*(Divide[z - Subscript[z, n],\[CapitalDelta]*z]+Divide[2*x,\[CapitalDelta]*x])]*(2*Exp[Divide[- 6*Pi*I*x,\[CapitalDelta]*x]]*Cos[Divide[2*Sqrt[3]*Pi*y,\[CapitalDelta]*x]]+ 1)=Sqrt[3] Failure Failure
Fail
-1.734788795+.1075442147e-1*I <- {Delta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), z[n] = 2^(1/2)+I*2^(1/2), x = 1, y = 1}
-1.766963148+.2014376123e-1*I <- {Delta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), z[n] = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
-.333546281+.6639225830*I <- {Delta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), z[n] = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
-1.735280293+.1131186187e-1*I <- {Delta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), z[n] = 2^(1/2)+I*2^(1/2), x = 2, y = 1}
... skip entries to safe data
 Skip
36.9.E2 ( Ai ⁑ ( x ) ) 2 = 2 2 / 3 Ο€ ⁒ ∫ 0 ∞ Ai ⁑ ( 2 2 / 3 ⁒ ( u 2 + x ) ) ⁒ d u superscript Airy-Ai π‘₯ 2 superscript 2 2 3 πœ‹ superscript subscript 0 Airy-Ai superscript 2 2 3 superscript 𝑒 2 π‘₯ 𝑒 {\displaystyle{\displaystyle(\mathrm{Ai}\left(x\right))^{2}=\frac{2^{2/3}}{\pi% }\int_{0}^{\infty}\mathrm{Ai}\left(2^{2/3}(u^{2}+x)\right)\mathrm{d}u}} (AiryAi(x))^(2)=((2)^(2/ 3))/(Pi)*int(AiryAi((2)^(2/ 3)*((u)^(2)+ x)), u = 0..infinity) (AiryAi[x])^(2)=Divide[(2)^(2/ 3),Pi]*Integrate[AiryAi[(2)^(2/ 3)*((u)^(2)+ x)], {u, 0, Infinity}] Failure Failure Skip Skip
36.12.E9 P m ⁒ n ⁒ ( 𝐲 ) = ( t n ⁒ ( 𝐱 ⁒ ( 𝐲 ) ) ) K + 1 + βˆ‘ l = m + 2 K l K + 2 ⁒ x l ⁒ ( 𝐲 ) ⁒ ( t n ⁒ ( 𝐱 ⁒ ( 𝐲 ) ) ) l - 1 subscript 𝑃 π‘š 𝑛 𝐲 superscript subscript 𝑑 𝑛 𝐱 𝐲 𝐾 1 superscript subscript 𝑙 π‘š 2 𝐾 𝑙 𝐾 2 subscript π‘₯ 𝑙 𝐲 superscript subscript 𝑑 𝑛 𝐱 𝐲 𝑙 1 {\displaystyle{\displaystyle P_{mn}(\mathbf{y})=(t_{n}(\mathbf{x}(\mathbf{y}))% )^{K+1}+\sum_{l=m+2}^{K}\frac{l}{K+2}x_{l}(\mathbf{y})(t_{n}(\mathbf{x}(% \mathbf{y})))^{l-1}}} P[m*n]*(y)=(t[n]*(x*(y)))^(K + 1)+ sum((l)/(K + 2)*x[l]*(y)*(t[n]*(x*(y)))^(l - 1), l = m + 2..K) Subscript[P, m*n]*(y)=(Subscript[t, n]*(x*(y)))^(K + 1)+ Sum[Divide[l,K + 2]*Subscript[x, l]*(y)*(Subscript[t, n]*(x*(y)))^(l - 1), {l, m + 2, K}] Failure Failure Skip Skip
36.13.E1 z ⁒ ( Ο• , ρ ) = ∫ - Ο€ / 2 Ο€ / 2 cos ⁑ ( ρ ⁒ cos ⁑ ( ΞΈ + Ο• ) cos 2 ⁑ ΞΈ ) ⁒ d ΞΈ 𝑧 italic-Ο• 𝜌 superscript subscript πœ‹ 2 πœ‹ 2 𝜌 πœƒ italic-Ο• 2 πœƒ πœƒ {\displaystyle{\displaystyle z(\phi,\rho)=\int_{-\pi/2}^{\pi/2}\cos\left(\rho% \frac{\cos\left(\theta+\phi\right)}{{\cos^{2}}\theta}\right)\mathrm{d}\theta}} z*(phi , rho)= int(cos(rho*(cos(theta + phi))/((cos(theta))^(2))), theta = - Pi/ 2..Pi/ 2) z*(\[Phi], \[Rho])= Integrate[Cos[\[Rho]*Divide[Cos[\[Theta]+ \[Phi]],(Cos[\[Theta]])^(2)]], {\[Theta], - Pi/ 2, Pi/ 2}] Failure Failure Skip Error
36.13#Ex1 ΞΈ + ⁒ ( Ο• ) = 1 2 ⁒ ( arcsin ⁑ ( 3 ⁒ sin ⁑ Ο• ) - Ο• ) subscript πœƒ italic-Ο• 1 2 3 italic-Ο• italic-Ο• {\displaystyle{\displaystyle\theta_{+}(\phi)=\tfrac{1}{2}(\operatorname{arcsin% }\left(3\sin\phi\right)-\phi)}} theta[+]*(phi)=(1)/(2)*(arcsin(3*sin(phi))- phi) Subscript[\[Theta], +]*(\[Phi])=Divide[1,2]*(ArcSin[3*Sin[\[Phi]]]- \[Phi]) Error Failure - Error
36.13#Ex2 ΞΈ - ⁒ ( Ο• ) = 1 2 ⁒ ( Ο€ - Ο• - arcsin ⁑ ( 3 ⁒ sin ⁑ Ο• ) ) subscript πœƒ italic-Ο• 1 2 πœ‹ italic-Ο• 3 italic-Ο• {\displaystyle{\displaystyle\theta_{-}(\phi)=\tfrac{1}{2}(\pi-\phi-% \operatorname{arcsin}\left(3\sin\phi\right))}} theta[-]*(phi)=(1)/(2)*(Pi - phi - arcsin(3*sin(phi))) Subscript[\[Theta], -]*(\[Phi])=Divide[1,2]*(Pi - \[Phi]- ArcSin[3*Sin[\[Phi]]]) Error Failure - Error
36.13.E5 | Ο• | = Ο• c italic-Ο• subscript italic-Ο• 𝑐 {\displaystyle{\displaystyle|\phi|=\phi_{c}}} abs(phi)= phi[c] Abs[\[Phi]]= Subscript[\[Phi], c] Failure Failure
Fail
.585786438-1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), phi[c] = 2^(1/2)+I*2^(1/2)}
.585786438+1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), phi[c] = 2^(1/2)-I*2^(1/2)}
3.414213562+1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), phi[c] = -2^(1/2)-I*2^(1/2)}
3.414213562-1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), phi[c] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Successful
36.13.E5 Ο• c = arcsin ⁑ ( 1 3 ) subscript italic-Ο• 𝑐 1 3 {\displaystyle{\displaystyle\phi_{c}=\operatorname{arcsin}\left(\tfrac{1}{3}% \right)}} phi[c]= arcsin((1)/(3)) Subscript[\[Phi], c]= ArcSin[Divide[1,3]] Failure Failure
Fail
1.074376653+1.414213562*I <- {phi[c] = 2^(1/2)+I*2^(1/2)}
1.074376653-1.414213562*I <- {phi[c] = 2^(1/2)-I*2^(1/2)}
-1.754050471-1.414213562*I <- {phi[c] = -2^(1/2)-I*2^(1/2)}
-1.754050471+1.414213562*I <- {phi[c] = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.074376652918973, 1.4142135623730951] <- {Rule[Subscript[Ο•, c], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.074376652918973, -1.4142135623730951] <- {Rule[Subscript[Ο•, c], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7540504718272172, -1.4142135623730951] <- {Rule[Subscript[Ο•, c], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7540504718272172, 1.4142135623730951] <- {Rule[Subscript[Ο•, c], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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