Results of Painlevé Transcendents

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
32.2.E1 d 2 w d z 2 = 6 ⁢ w 2 + z derivative 𝑤 𝑧 2 6 superscript 𝑤 2 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=6w^{2}% +z}} diff(w, [z$(2)])= 6*(w)^(2)+ z D[w, {z, 2}]= 6*(w)^(2)+ z Failure Failure
Fail
-1.414213562-25.41421355*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1.414213562-22.58578643*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.414213562-22.58578643*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.414213562-25.41421355*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-1.4142135623730951, -25.414213562373096] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -22.585786437626904] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -22.585786437626904] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -25.414213562373096] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
32.2.E2 d 2 w d z 2 = 2 ⁢ w 3 + z ⁢ w + α derivative 𝑤 𝑧 2 2 superscript 𝑤 3 𝑧 𝑤 𝛼 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=2w^{3}% +zw+\alpha}} diff(w, [z$(2)])= 2*(w)^(3)+ z*w + alpha D[w, {z, 2}]= 2*(w)^(3)+ z*w + \[Alpha] Failure Failure
Fail
9.899494928-16.72792205*I <- {alpha = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
5.899494930-12.72792205*I <- {alpha = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
9.899494928-8.727922054*I <- {alpha = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
13.89949493-12.72792205*I <- {alpha = 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[9.899494936611665, -16.72792206135786] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[9.899494936611665, -13.899494936611665] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[12.727922061357857, -13.899494936611665] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[12.727922061357857, -16.72792206135786] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
32.2.E3 d 2 w d z 2 = 1 w ⁢ ( d w d z ) 2 - 1 z ⁢ d w d z + α ⁢ w 2 + β z + γ ⁢ w 3 + δ w derivative 𝑤 𝑧 2 1 𝑤 superscript derivative 𝑤 𝑧 2 1 𝑧 derivative 𝑤 𝑧 𝛼 superscript 𝑤 2 𝛽 𝑧 𝛾 superscript 𝑤 3 𝛿 𝑤 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{% 1}{w}\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-\frac{1}{z}\frac{\mathrm% {d}w}{\mathrm{d}z}+\frac{\alpha w^{2}+\beta}{z}+\gamma w^{3}+\frac{\delta}{w}}} diff(w, [z$(2)])=(1)/(w)*(diff(w, z))^(2)-(1)/(z)*diff(w, z)+(alpha*(w)^(2)+ beta)/(z)+ gamma*(w)^(3)+(delta)/(w) D[w, {z, 2}]=Divide[1,w]*(D[w, z])^(2)-Divide[1,z]*D[w, z]+Divide[\[Alpha]*(w)^(2)+ \[Beta],z]+ \[Gamma]*(w)^(3)+Divide[\[Delta],w] Failure Failure Skip Skip
32.2.E4 d 2 w d z 2 = 1 2 ⁢ w ⁢ ( d w d z ) 2 + 3 2 ⁢ w 3 + 4 ⁢ z ⁢ w 2 + 2 ⁢ ( z 2 - α ) ⁢ w + β w derivative 𝑤 𝑧 2 1 2 𝑤 superscript derivative 𝑤 𝑧 2 3 2 superscript 𝑤 3 4 𝑧 superscript 𝑤 2 2 superscript 𝑧 2 𝛼 𝑤 𝛽 𝑤 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{% 1}{2w}\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+\frac{3}{2}w^{3}+4zw^{2% }+2(z^{2}-\alpha)w+\frac{\beta}{w}}} diff(w, [z$(2)])=(1)/(2*w)*(diff(w, z))^(2)+(3)/(2)*(w)^(3)+ 4*z*(w)^(2)+ 2*((z)^(2)- alpha)* w +(beta)/(w) D[w, {z, 2}]=Divide[1,2*w]*(D[w, z])^(2)+Divide[3,2]*(w)^(3)+ 4*z*(w)^(2)+ 2*((z)^(2)- \[Alpha])* w +Divide[\[Beta],w] Failure Failure
Fail
41.42640684-34.42640684*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-26.45584410-11.79898986*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-3.82842712+10.82842712*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
18.79898986+33.45584410*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 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[41.42640687119285, -34.42640687119285] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[β, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[42.42640687119285, -33.42640687119285] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[β, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[43.42640687119285, -34.42640687119285] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[β, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[42.42640687119285, -35.42640687119285] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[β, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
32.2.E5 d 2 w d z 2 = ( 1 2 ⁢ w + 1 w - 1 ) ⁢ ( d w d z ) 2 - 1 z ⁢ d w d z + ( w - 1 ) 2 z 2 ⁢ ( α ⁢ w + β w ) + γ ⁢ w z + δ ⁢ w ⁢ ( w + 1 ) w - 1 derivative 𝑤 𝑧 2 1 2 𝑤 1 𝑤 1 superscript derivative 𝑤 𝑧 2 1 𝑧 derivative 𝑤 𝑧 superscript 𝑤 1 2 superscript 𝑧 2 𝛼 𝑤 𝛽 𝑤 𝛾 𝑤 𝑧 𝛿 𝑤 𝑤 1 𝑤 1 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\left(% \frac{1}{2w}+\frac{1}{w-1}\right)\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^% {2}-\frac{1}{z}\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{(w-1)^{2}}{z^{2}}\left(% \alpha w+\frac{\beta}{w}\right)+\frac{\gamma w}{z}+\frac{\delta w(w+1)}{w-1}}} diff(w, [z$(2)])=((1)/(2*w)+(1)/(w - 1))*(diff(w, z))^(2)-(1)/(z)*diff(w, z)+((w - 1)^(2))/((z)^(2))*(alpha*w +(beta)/(w))+(gamma*w)/(z)+(delta*w*(w + 1))/(w - 1) D[w, {z, 2}]=(Divide[1,2*w]+Divide[1,w - 1])*(D[w, z])^(2)-Divide[1,z]*D[w, z]+Divide[(w - 1)^(2),(z)^(2)]*(\[Alpha]*w +Divide[\[Beta],w])+Divide[\[Gamma]*w,z]+Divide[\[Delta]*w*(w + 1),w - 1] Failure Failure Skip Skip
32.2.E6 d 2 w d z 2 = 1 2 ⁢ ( 1 w + 1 w - 1 + 1 w - z ) ⁢ ( d w d z ) 2 - ( 1 z + 1 z - 1 + 1 w - z ) ⁢ d w d z + w ⁢ ( w - 1 ) ⁢ ( w - z ) z 2 ⁢ ( z - 1 ) 2 ⁢ ( α + β ⁢ z w 2 + γ ⁢ ( z - 1 ) ( w - 1 ) 2 + δ ⁢ z ⁢ ( z - 1 ) ( w - z ) 2 ) derivative 𝑤 𝑧 2 1 2 1 𝑤 1 𝑤 1 1 𝑤 𝑧 superscript derivative 𝑤 𝑧 2 1 𝑧 1 𝑧 1 1 𝑤 𝑧 derivative 𝑤 𝑧 𝑤 𝑤 1 𝑤 𝑧 superscript 𝑧 2 superscript 𝑧 1 2 𝛼 𝛽 𝑧 superscript 𝑤 2 𝛾 𝑧 1 superscript 𝑤 1 2 𝛿 𝑧 𝑧 1 superscript 𝑤 𝑧 2 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{% 1}{2}\left(\frac{1}{w}+\frac{1}{w-1}+\frac{1}{w-z}\right)\left(\frac{\mathrm{d% }w}{\mathrm{d}z}\right)^{2}-\left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{w-z}% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{w(w-1)(w-z)}{z^{2}(z-1)^{2}}\left% (\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+\frac{\delta z(z-1% )}{(w-z)^{2}}\right)}} diff(w, [z$(2)])=(1)/(2)*((1)/(w)+(1)/(w - 1)+(1)/(w - z))*(diff(w, z))^(2)-((1)/(z)+(1)/(z - 1)+(1)/(w - z))* diff(w, z)+(w*(w - 1)*(w - z))/((z)^(2)*(z - 1)^(2))*(alpha +(beta*z)/((w)^(2))+(gamma*(z - 1))/((w - 1)^(2))+(delta*z*(z - 1))/((w - z)^(2))) D[w, {z, 2}]=Divide[1,2]*(Divide[1,w]+Divide[1,w - 1]+Divide[1,w - z])*(D[w, z])^(2)-(Divide[1,z]+Divide[1,z - 1]+Divide[1,w - z])* D[w, z]+Divide[w*(w - 1)*(w - z),(z)^(2)*(z - 1)^(2)]*(\[Alpha]+Divide[\[Beta]*z,(w)^(2)]+Divide[\[Gamma]*(z - 1),(w - 1)^(2)]+Divide[\[Delta]*z*(z - 1),(w - z)^(2)]) Failure Failure Skip Skip
32.2.E7 d 2 w d z 2 = F ⁢ ( z , w , d w d z ) derivative 𝑤 𝑧 2 𝐹 𝑧 𝑤 derivative 𝑤 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=F\left% (z,w,\frac{\mathrm{d}w}{\mathrm{d}z}\right)}} diff(w, [z$(2)])= F*(z , w , diff(w, z)) D[w, {z, 2}]= F*(z , w , D[w, z]) Failure Failure Error Error
32.2.E9 d 2 u d ζ 2 = 1 u ⁢ ( d u d ζ ) 2 - 1 ζ ⁢ d u d ζ + u 2 ⁢ ( α + γ ⁢ u ) 4 ⁢ ζ 2 + β 4 ⁢ ζ + δ 4 ⁢ u derivative 𝑢 𝜁 2 1 𝑢 superscript derivative 𝑢 𝜁 2 1 𝜁 derivative 𝑢 𝜁 superscript 𝑢 2 𝛼 𝛾 𝑢 4 superscript 𝜁 2 𝛽 4 𝜁 𝛿 4 𝑢 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=% \frac{1}{u}\left(\frac{\mathrm{d}u}{\mathrm{d}\zeta}\right)^{2}-\frac{1}{\zeta% }\frac{\mathrm{d}u}{\mathrm{d}\zeta}+\frac{u^{2}(\alpha+\gamma u)}{4\zeta^{2}}% +\frac{\beta}{4\zeta}+\frac{\delta}{4u}}} diff(u, [zeta$(2)])=(1)/(u)*(diff(u, zeta))^(2)-(1)/(zeta)*diff(u, zeta)+((u)^(2)*(alpha + gamma*u))/(4*(zeta)^(2))+(beta)/(4*zeta)+(delta)/(4*u) D[u, {\[zeta], 2}]=Divide[1,u]*(D[u, \[zeta]])^(2)-Divide[1,\[zeta]]*D[u, \[zeta]]+Divide[(u)^(2)*(\[Alpha]+ \[Gamma]*u),4*(\[zeta])^(2)]+Divide[\[Beta],4*\[zeta]]+Divide[\[Delta],4*u] Failure Failure Skip Error
32.2.E10 d 2 u d z 2 + 1 z ⁢ d u d z = 2 ⁢ α z ⁢ sin ⁡ u + 2 ⁢ γ ⁢ sin ⁡ ( 2 ⁢ u ) derivative 𝑢 𝑧 2 1 𝑧 derivative 𝑢 𝑧 2 𝛼 𝑧 𝑢 2 𝛾 2 𝑢 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}z}^{2}}+\frac{% 1}{z}\frac{\mathrm{d}u}{\mathrm{d}z}=\frac{2\alpha}{z}\sin u+2\gamma\sin\left(% 2u\right)}} diff(u, [z$(2)])+(1)/(z)*diff(u, z)=(2*alpha)/(z)*sin(u)+ 2*gamma*sin(2*u) D[u, {z, 2}]+Divide[1,z]*D[u, z]=Divide[2*\[Alpha],z]*Sin[u]+ 2*\[Gamma]*Sin[2*u] Failure Failure
Fail
-7.322152505+8.654853973*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-2.415558484+4.955305834*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.283989655+9.861899855*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-3.622604366+13.56144799*I <- {alpha = 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
32.2.E11 d 2 u d ζ 2 = 3 ⁢ u 5 + 2 ⁢ ζ ⁢ u 3 + ( 1 4 ⁢ ζ 2 - ν - 1 2 ) ⁢ u + β 32 ⁢ u 3 derivative 𝑢 𝜁 2 3 superscript 𝑢 5 2 𝜁 superscript 𝑢 3 1 4 superscript 𝜁 2 𝜈 1 2 𝑢 𝛽 32 superscript 𝑢 3 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=3u% ^{5}+2\zeta u^{3}+\left(\tfrac{1}{4}\zeta^{2}-\nu-\tfrac{1}{2}\right)u+\frac{% \beta}{32u^{3}}}} diff(u, [zeta$(2)])= 3*(u)^(5)+ 2*zeta*(u)^(3)+((1)/(4)*(zeta)^(2)- nu -(1)/(2))* u +(beta)/(32*(u)^(3)) D[u, {\[zeta], 2}]= 3*(u)^(5)+ 2*\[zeta]*(u)^(3)+(Divide[1,4]*(\[zeta])^(2)- \[Nu]-Divide[1,2])* u +Divide[\[Beta],32*(u)^(3)] Failure Failure
Fail
102.0035712+71.18295663*I <- {beta = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
67.17514413+42.01138379*I <- {beta = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
38.00357129+71.18295663*I <- {beta = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
67.17514413+106.0113837*I <- {beta = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.2.E12 d 2 u d ζ 2 = - α ⁢ cosh ⁡ u 2 ⁢ ( sinh ⁡ u ) 3 - β ⁢ sinh ⁡ u 2 ⁢ ( cosh ⁡ u ) 3 - 1 4 ⁢ γ ⁢ e ζ ⁢ sinh ⁡ ( 2 ⁢ u ) - 1 8 ⁢ δ ⁢ e 2 ⁢ ζ ⁢ sinh ⁡ ( 4 ⁢ u ) derivative 𝑢 𝜁 2 𝛼 𝑢 2 superscript 𝑢 3 𝛽 𝑢 2 superscript 𝑢 3 1 4 𝛾 superscript 𝑒 𝜁 2 𝑢 1 8 𝛿 superscript 𝑒 2 𝜁 4 𝑢 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=-% \frac{\alpha\cosh u}{2(\sinh u)^{3}}-\frac{\beta\sinh u}{2(\cosh u)^{3}}-% \tfrac{1}{4}\gamma e^{\zeta}\sinh\left(2u\right)-\tfrac{1}{8}\delta e^{2\zeta}% \sinh\left(4u\right)}} diff(u, [zeta$(2)])= -(alpha*cosh(u))/(2*(sinh(u))^(3))-(beta*sinh(u))/(2*(cosh(u))^(3))-(1)/(4)*gamma*exp(zeta)*sinh(2*u)-(1)/(8)*delta*exp(2*zeta)*sinh(4*u) D[u, {\[zeta], 2}]= -Divide[\[Alpha]*Cosh[u],2*(Sinh[u])^(3)]-Divide[\[Beta]*Sinh[u],2*(Cosh[u])^(3)]-Divide[1,4]*\[Gamma]*Exp[\[zeta]]*Sinh[2*u]-Divide[1,8]*\[Delta]*Exp[2*\[zeta]]*Sinh[4*u] Failure Failure Skip Error
32.2.E13 z ⁢ ( 1 - z ) ⁢ I ⁢ ( ∫ ∞ w d t t ⁢ ( t - 1 ) ⁢ ( t - z ) ) = w ⁢ ( w - 1 ) ⁢ ( w - z ) ⁢ ( α + β ⁢ z w 2 + γ ⁢ ( z - 1 ) ( w - 1 ) 2 + ( δ - 1 2 ) ⁢ z ⁢ ( z - 1 ) ( w - z ) 2 ) 𝑧 1 𝑧 𝐼 superscript subscript 𝑤 𝑡 𝑡 𝑡 1 𝑡 𝑧 𝑤 𝑤 1 𝑤 𝑧 𝛼 𝛽 𝑧 superscript 𝑤 2 𝛾 𝑧 1 superscript 𝑤 1 2 𝛿 1 2 𝑧 𝑧 1 superscript 𝑤 𝑧 2 {\displaystyle{\displaystyle z(1-z)I\left(\int_{\infty}^{w}\frac{\mathrm{d}t}{% \sqrt{t(t-1)(t-z)}}\right)=\sqrt{w(w-1)(w-z)}\*\left(\alpha+\frac{\beta z}{w^{% 2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+(\delta-\tfrac{1}{2})\frac{z(z-1)}{(w-z)^{2}% }\right)}} z*(1 - z)* I*(int((1)/(sqrt(t*(t - 1)*(t - z))), t = infinity..w))=sqrt(w*(w - 1)*(w - z))*(alpha +(beta*z)/((w)^(2))+(gamma*(z - 1))/((w - 1)^(2))+(delta -(1)/(2))*(z*(z - 1))/((w - z)^(2))) z*(1 - z)* I*(Integrate[Divide[1,Sqrt[t*(t - 1)*(t - z)]], {t, Infinity, w}])=Sqrt[w*(w - 1)*(w - z)]*(\[Alpha]+Divide[\[Beta]*z,(w)^(2)]+Divide[\[Gamma]*(z - 1),(w - 1)^(2)]+(\[Delta]-Divide[1,2])*Divide[z*(z - 1),(w - z)^(2)]) Failure Failure Skip Error
32.2.E14 I = z ⁢ ( 1 - z ) ⁢ d 2 d z 2 + ( 1 - 2 ⁢ z ) ⁢ d d z - 1 4 𝐼 𝑧 1 𝑧 derivative 𝑧 2 1 2 𝑧 derivative 𝑧 1 4 {\displaystyle{\displaystyle I=z(1-z)\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}% }+(1-2z)\frac{\mathrm{d}}{\mathrm{d}z}-\frac{1}{4}}} I = z*(1 - z)* diff(+(1 - 2*z)* diff(-, z), [z$(2)])(1)/(4) I = z*(1 - z)* D[+(1 - 2*z)* D[-, z], {z, 2}]Divide[1,4] Error Failure - Error
32.2#Ex3 d f 1 d z + f 1 ⁢ ( f 2 - f 3 ) + 2 ⁢ μ 1 = 0 derivative subscript 𝑓 1 𝑧 subscript 𝑓 1 subscript 𝑓 2 subscript 𝑓 3 2 subscript 𝜇 1 0 {\displaystyle{\displaystyle\frac{\mathrm{d}f_{1}}{\mathrm{d}z}+f_{1}(f_{2}-f_% {3})+2\mu_{1}=0}} diff(f[1], z)+ f[1]*(f[2]- f[3])+ 2*mu[1]= 0 D[Subscript[f, 1], z]+ Subscript[f, 1]*(Subscript[f, 2]- Subscript[f, 3])+ 2*Subscript[\[Mu], 1]= 0 Failure Failure
Fail
2.828427124+2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[1] = 2^(1/2)+I*2^(1/2)}
2.828427124-2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[1] = 2^(1/2)-I*2^(1/2)}
-2.828427124-2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[1] = -2^(1/2)-I*2^(1/2)}
-2.828427124+2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[1] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, -2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, 2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
32.2#Ex4 d f 2 d z + f 2 ⁢ ( f 3 - f 1 ) + 2 ⁢ μ 2 = 0 derivative subscript 𝑓 2 𝑧 subscript 𝑓 2 subscript 𝑓 3 subscript 𝑓 1 2 subscript 𝜇 2 0 {\displaystyle{\displaystyle\frac{\mathrm{d}f_{2}}{\mathrm{d}z}+f_{2}(f_{3}-f_% {1})+2\mu_{2}=0}} diff(f[2], z)+ f[2]*(f[3]- f[1])+ 2*mu[2]= 0 D[Subscript[f, 2], z]+ Subscript[f, 2]*(Subscript[f, 3]- Subscript[f, 1])+ 2*Subscript[\[Mu], 2]= 0 Failure Failure
Fail
2.828427124+2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[2] = 2^(1/2)+I*2^(1/2)}
2.828427124-2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[2] = 2^(1/2)-I*2^(1/2)}
-2.828427124-2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[2] = -2^(1/2)-I*2^(1/2)}
-2.828427124+2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[2] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
32.2#Ex5 d f 3 d z + f 3 ⁢ ( f 1 - f 2 ) + 2 ⁢ μ 3 = 0 derivative subscript 𝑓 3 𝑧 subscript 𝑓 3 subscript 𝑓 1 subscript 𝑓 2 2 subscript 𝜇 3 0 {\displaystyle{\displaystyle\frac{\mathrm{d}f_{3}}{\mathrm{d}z}+f_{3}(f_{1}-f_% {2})+2\mu_{3}=0}} diff(f[3], z)+ f[3]*(f[1]- f[2])+ 2*mu[3]= 0 D[Subscript[f, 3], z]+ Subscript[f, 3]*(Subscript[f, 1]- Subscript[f, 2])+ 2*Subscript[\[Mu], 3]= 0 Failure Failure
Fail
2.828427124+2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[3] = 2^(1/2)+I*2^(1/2)}
2.828427124-2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[3] = 2^(1/2)-I*2^(1/2)}
-2.828427124-2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[3] = -2^(1/2)-I*2^(1/2)}
-2.828427124+2.828427124*I <- {f[1] = 2^(1/2)+I*2^(1/2), f[2] = 2^(1/2)+I*2^(1/2), f[3] = 2^(1/2)+I*2^(1/2), mu[3] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, -2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 3], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, 2.8284271247461903] <- {Rule[Subscript[f, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[f, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[μ, 3], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
32.2#Ex6 z ⁢ d f 1 d z = f 1 ⁢ f 3 ⁢ ( f 2 - f 4 ) + ( 1 2 - μ 3 ) ⁢ f 1 + μ 1 ⁢ f 3 𝑧 derivative subscript 𝑓 1 𝑧 subscript 𝑓 1 subscript 𝑓 3 subscript 𝑓 2 subscript 𝑓 4 1 2 subscript 𝜇 3 subscript 𝑓 1 subscript 𝜇 1 subscript 𝑓 3 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{1}}{\mathrm{d}z}=f_{1}f_{3}(f% _{2}-f_{4})+(\tfrac{1}{2}-\mu_{3})f_{1}+\mu_{1}f_{3}}} z*diff(f[1], z)= f[1]*f[3]*(f[2]- f[4])+((1)/(2)- mu[3])* f[1]+ mu[1]*f[3] z*D[Subscript[f, 1], z]= Subscript[f, 1]*Subscript[f, 3]*(Subscript[f, 2]- Subscript[f, 4])+(Divide[1,2]- Subscript[\[Mu], 3])* Subscript[f, 1]+ Subscript[\[Mu], 1]*Subscript[f, 3] Failure Failure Skip Skip
32.2#Ex7 z ⁢ d f 2 d z = f 2 ⁢ f 4 ⁢ ( f 3 - f 1 ) + ( 1 2 - μ 4 ) ⁢ f 2 + μ 2 ⁢ f 4 𝑧 derivative subscript 𝑓 2 𝑧 subscript 𝑓 2 subscript 𝑓 4 subscript 𝑓 3 subscript 𝑓 1 1 2 subscript 𝜇 4 subscript 𝑓 2 subscript 𝜇 2 subscript 𝑓 4 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{2}}{\mathrm{d}z}=f_{2}f_{4}(f% _{3}-f_{1})+(\tfrac{1}{2}-\mu_{4})f_{2}+\mu_{2}f_{4}}} z*diff(f[2], z)= f[2]*f[4]*(f[3]- f[1])+((1)/(2)- mu[4])* f[2]+ mu[2]*f[4] z*D[Subscript[f, 2], z]= Subscript[f, 2]*Subscript[f, 4]*(Subscript[f, 3]- Subscript[f, 1])+(Divide[1,2]- Subscript[\[Mu], 4])* Subscript[f, 2]+ Subscript[\[Mu], 2]*Subscript[f, 4] Failure Failure Skip Skip
32.2#Ex8 z ⁢ d f 3 d z = f 3 ⁢ f 1 ⁢ ( f 4 - f 2 ) + ( 1 2 - μ 1 ) ⁢ f 3 + μ 3 ⁢ f 1 𝑧 derivative subscript 𝑓 3 𝑧 subscript 𝑓 3 subscript 𝑓 1 subscript 𝑓 4 subscript 𝑓 2 1 2 subscript 𝜇 1 subscript 𝑓 3 subscript 𝜇 3 subscript 𝑓 1 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{3}}{\mathrm{d}z}=f_{3}f_{1}(f% _{4}-f_{2})+(\tfrac{1}{2}-\mu_{1})f_{3}+\mu_{3}f_{1}}} z*diff(f[3], z)= f[3]*f[1]*(f[4]- f[2])+((1)/(2)- mu[1])* f[3]+ mu[3]*f[1] z*D[Subscript[f, 3], z]= Subscript[f, 3]*Subscript[f, 1]*(Subscript[f, 4]- Subscript[f, 2])+(Divide[1,2]- Subscript[\[Mu], 1])* Subscript[f, 3]+ Subscript[\[Mu], 3]*Subscript[f, 1] Failure Failure Skip Skip
32.2#Ex9 z ⁢ d f 4 d z = f 4 ⁢ f 2 ⁢ ( f 1 - f 3 ) + ( 1 2 - μ 2 ) ⁢ f 4 + μ 4 ⁢ f 2 𝑧 derivative subscript 𝑓 4 𝑧 subscript 𝑓 4 subscript 𝑓 2 subscript 𝑓 1 subscript 𝑓 3 1 2 subscript 𝜇 2 subscript 𝑓 4 subscript 𝜇 4 subscript 𝑓 2 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{4}}{\mathrm{d}z}=f_{4}f_{2}(f% _{1}-f_{3})+(\tfrac{1}{2}-\mu_{2})f_{4}+\mu_{4}f_{2}}} z*diff(f[4], z)= f[4]*f[2]*(f[1]- f[3])+((1)/(2)- mu[2])* f[4]+ mu[4]*f[2] z*D[Subscript[f, 4], z]= Subscript[f, 4]*Subscript[f, 2]*(Subscript[f, 1]- Subscript[f, 3])+(Divide[1,2]- Subscript[\[Mu], 2])* Subscript[f, 4]+ Subscript[\[Mu], 4]*Subscript[f, 2] Failure Failure Skip Skip
32.2.E27 d 2 W d ζ 2 = 6 ⁢ W 2 + ζ + ϵ 6 ⁢ ( 2 ⁢ W 3 + ζ ⁢ W ) derivative 𝑊 𝜁 2 6 superscript 𝑊 2 𝜁 superscript italic-ϵ 6 2 superscript 𝑊 3 𝜁 𝑊 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\zeta}^{2}}=6W% ^{2}+\zeta+\epsilon^{6}(2W^{3}+\zeta W)}} diff(W, [zeta$(2)])= 6*(W)^(2)+ zeta + (epsilon)^(6)*(2*(W)^(3)+ zeta*W) D[W, {\[zeta], 2}]= 6*(W)^(2)+ \[zeta]+ (\[Epsilon])^(6)*(2*(W)^(3)+ \[zeta]*W) Failure Failure
Fail
-981.4915554-749.4915558*I <- {W = 2^(1/2)+I*2^(1/2), epsilon = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
-725.4915558-490.6631292*I <- {W = 2^(1/2)+I*2^(1/2), epsilon = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
-466.6631292-746.6631286*I <- {W = 2^(1/2)+I*2^(1/2), epsilon = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
-722.6631286-1005.491555*I <- {W = 2^(1/2)+I*2^(1/2), epsilon = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.3.E2 d 2 u d x 2 = 3 ⁢ u 5 + 2 ⁢ x ⁢ u 3 + ( 1 4 ⁢ x 2 - ν - 1 2 ) ⁢ u derivative 𝑢 𝑥 2 3 superscript 𝑢 5 2 𝑥 superscript 𝑢 3 1 4 superscript 𝑥 2 𝜈 1 2 𝑢 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}x}^{2}}=3u^{5}% +2xu^{3}+\left(\tfrac{1}{4}x^{2}-\nu-\tfrac{1}{2}\right)u}} diff(u, [x$(2)])= 3*(u)^(5)+ 2*x*(u)^(3)+((1)/(4)*(x)^(2)- nu -(1)/(2))* u D[u, {x, 2}]= 3*(u)^(5)+ 2*x*(u)^(3)+(Divide[1,4]*(x)^(2)- \[Nu]-Divide[1,2])* u Failure Failure
Fail
79.54951279+60.92209581*I <- {nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), x = 1}
89.80256111+48.54772715*I <- {nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), x = 2}
99.34850267+35.46625170*I <- {nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), x = 3}
83.54951279-56.92209581*I <- {nu = 2^(1/2)+I*2^(1/2), u = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[79.5495128834866, 60.92209588551708] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[89.80256121069154, 48.5477272147525] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[99.34850275670993, 35.46625176280137] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[83.5495128834866, 56.92209588551708] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
32.4#Ex1 ∂ ⁡ 𝚿 ∂ ⁡ λ = 𝐀 ⁢ ( z , λ ) ⁢ 𝚿 partial-derivative 𝚿 𝜆 𝐀 𝑧 𝜆 𝚿 {\displaystyle{\displaystyle\frac{\partial\boldsymbol{{\Psi}}}{\partial\lambda% }=\mathbf{A}(z,\lambda)\boldsymbol{{\Psi}}}} diff(Psi, lambda)= A*(z , lambda)* Psi D[\[CapitalPsi], \[Lambda]]= A*(z , \[Lambda])* \[CapitalPsi] Failure Failure Error Error
32.4#Ex2 ∂ ⁡ 𝚿 ∂ ⁡ z = 𝐁 ⁢ ( z , λ ) ⁢ 𝚿 partial-derivative 𝚿 𝑧 𝐁 𝑧 𝜆 𝚿 {\displaystyle{\displaystyle\frac{\partial\boldsymbol{{\Psi}}}{\partial z}=% \mathbf{B}(z,\lambda)\boldsymbol{{\Psi}}}} diff(Psi, z)= B*(z , lambda)* Psi D[\[CapitalPsi], z]= B*(z , \[Lambda])* \[CapitalPsi] Failure Failure Error Error
32.4.E3 ∂ ⁡ 𝐀 ∂ ⁡ z - ∂ ⁡ 𝐁 ∂ ⁡ λ + 𝐀𝐁 - 𝐁𝐀 = 0 partial-derivative 𝐀 𝑧 partial-derivative 𝐁 𝜆 𝐀𝐁 𝐁𝐀 0 {\displaystyle{\displaystyle\frac{\partial\mathbf{A}}{\partial z}-\frac{% \partial\mathbf{B}}{\partial\lambda}+\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}% =0}} diff(A, z)- diff(B, lambda)+ A*B - B*A = 0 D[A, z]- D[B, \[Lambda]]+ A*B - B*A = 0 Successful Successful - -
32.6#Ex1 d q d z = ∂ ⁡ H ∂ ⁡ p derivative 𝑞 𝑧 partial-derivative H 𝑝 {\displaystyle{\displaystyle\frac{\mathrm{d}q}{\mathrm{d}z}=\frac{\partial% \mathrm{H}}{\partial p}}} diff(q, z)= diff(H, p) D[q, z]= D[H, p] Successful Successful - -
32.6#Ex2 d p d z = - ∂ ⁡ H ∂ ⁡ q derivative 𝑝 𝑧 partial-derivative H 𝑞 {\displaystyle{\displaystyle\frac{\mathrm{d}p}{\mathrm{d}z}=-\frac{\partial% \mathrm{H}}{\partial q}}} diff(p, z)= - diff(H, q) D[p, z]= - D[H, q] Successful Successful - -
32.8.E8 ∑ m = 0 ∞ p m ⁢ ( z ) ⁢ λ m = exp ⁡ ( z ⁢ λ - 4 3 ⁢ λ 3 ) superscript subscript 𝑚 0 subscript 𝑝 𝑚 𝑧 superscript 𝜆 𝑚 𝑧 𝜆 4 3 superscript 𝜆 3 {\displaystyle{\displaystyle\sum_{m=0}^{\infty}p_{m}(z)\lambda^{m}=\exp\left(z% \lambda-\tfrac{4}{3}\lambda^{3}\right)}} sum(p[m]*(z)* (lambda)^(m), m = 0..infinity)= exp(z*lambda -(4)/(3)*(lambda)^(3)) Sum[Subscript[p, m]*(z)* (\[Lambda])^(m), {m, 0, Infinity}]= Exp[z*\[Lambda]-Divide[4,3]*(\[Lambda])^(3)] Failure Failure Skip Skip
32.10.E5 ϕ ⁢ ( z ) = C 1 ⁢ Ai ⁡ ( - 2 - 1 / 3 ⁢ z ) + C 2 ⁢ Bi ⁡ ( - 2 - 1 / 3 ⁢ z ) italic-ϕ 𝑧 subscript 𝐶 1 Airy-Ai superscript 2 1 3 𝑧 subscript 𝐶 2 Airy-Bi superscript 2 1 3 𝑧 {\displaystyle{\displaystyle\phi(z)=C_{1}\mathrm{Ai}\left(-2^{-1/3}z\right)+C_% {2}\mathrm{Bi}\left(-2^{-1/3}z\right)}} phi*(z)= C[1]*AiryAi(- (2)^(- 1/ 3)* z)+ C[2]*AiryBi(- (2)^(- 1/ 3)* z) \[Phi]*(z)= Subscript[C, 1]*AiryAi[- (2)^(- 1/ 3)* z]+ Subscript[C, 2]*AiryBi[- (2)^(- 1/ 3)* z] Failure Failure
Fail
-2.625238972+3.502678312*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), C[1] = 2^(1/2)+I*2^(1/2), C[2] = 2^(1/2)+I*2^(1/2)}
-.2695111060+3.916247040*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), C[1] = 2^(1/2)+I*2^(1/2), C[2] = 2^(1/2)-I*2^(1/2)}
.144057622+1.560519174*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), C[1] = 2^(1/2)+I*2^(1/2), C[2] = -2^(1/2)-I*2^(1/2)}
-2.211670244+1.146950446*I <- {phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), C[1] = 2^(1/2)+I*2^(1/2), C[2] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
32.10.E14 ϕ ⁢ ( z ) = z ν ⁢ ( C 1 ⁢ J ν ⁡ ( ζ ) + C 2 ⁢ Y ν ⁡ ( ζ ) ) italic-ϕ 𝑧 superscript 𝑧 𝜈 subscript 𝐶 1 Bessel-J 𝜈 𝜁 subscript 𝐶 2 Bessel-Y-Weber 𝜈 𝜁 {\displaystyle{\displaystyle\phi(z)=z^{\nu}\left(C_{1}J_{\nu}\left(\zeta\right% )+C_{2}Y_{\nu}\left(\zeta\right)\right)}} phi*(z)= (z)^(nu)*(C[1]*BesselJ(nu, zeta)+ C[2]*BesselY(nu, zeta)) \[Phi]*(z)= (z)^(\[Nu])*(Subscript[C, 1]*BesselJ[\[Nu], \[zeta]]+ Subscript[C, 2]*BesselY[\[Nu], \[zeta]]) Failure Failure Skip Error
32.10.E19 ϕ ⁢ ( z ) = ( C 1 ⁢ U ⁡ ( a , 2 ⁢ z ) + C 2 ⁢ V ⁡ ( a , 2 ⁢ z ) ) ⁢ exp ⁡ ( 1 2 ⁢ ε ⁢ z 2 ) italic-ϕ 𝑧 subscript 𝐶 1 parabolic-U 𝑎 2 𝑧 subscript 𝐶 2 parabolic-V 𝑎 2 𝑧 1 2 𝜀 superscript 𝑧 2 {\displaystyle{\displaystyle\phi(z)=\left(C_{1}U\left(a,\sqrt{2}z\right)+C_{2}% V\left(a,\sqrt{2}z\right)\right)\exp\left(\tfrac{1}{2}\varepsilon z^{2}\right)}} phi*(z)=(C[1]*CylinderU(a, sqrt(2)*z)+ C[2]*CylinderV(a, sqrt(2)*z))* exp((1)/(2)*varepsilon*(z)^(2)) Error Failure Error Skip -
32.10.E27 ϕ ⁢ ( z ) = C 1 ⁢ M κ , μ ⁡ ( ζ ) + C 2 ⁢ W κ , μ ⁡ ( ζ ) ζ ( a - b + 1 ) / 2 ⁢ exp ⁡ ( 1 2 ⁢ ζ ) italic-ϕ 𝑧 subscript 𝐶 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝜁 subscript 𝐶 2 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝜁 superscript 𝜁 𝑎 𝑏 1 2 1 2 𝜁 {\displaystyle{\displaystyle\phi(z)=\frac{C_{1}M_{\kappa,\mu}\left(\zeta\right% )+C_{2}W_{\kappa,\mu}\left(\zeta\right)}{\zeta^{(a-b+1)/2}}\exp\left(\tfrac{1}% {2}\zeta\right)}} phi*(z)=(C[1]*WhittakerM(kappa, mu, zeta)+ C[2]*WhittakerW(kappa, mu, zeta))/((zeta)^((a - b + 1)/ 2))*exp((1)/(2)*zeta) \[Phi]*(z)=Divide[Subscript[C, 1]*WhittakerM[\[Kappa], \[Mu], \[zeta]]+ Subscript[C, 2]*WhittakerW[\[Kappa], \[Mu], \[zeta]],(\[zeta])^((a - b + 1)/ 2)]*Exp[Divide[1,2]*\[zeta]] Failure Failure Skip Error
32.10#Ex1 w ⁢ ( z ) = ζ - 1 a ⁢ ϕ ⁢ ( ζ ) ⁢ d ϕ d ζ 𝑤 𝑧 𝜁 1 𝑎 italic-ϕ 𝜁 derivative italic-ϕ 𝜁 {\displaystyle{\displaystyle w(z)=\frac{\zeta-1}{a\phi(\zeta)}\frac{\mathrm{d}% \phi}{\mathrm{d}\zeta}}} w*(z)=(zeta - 1)/(a*phi*(zeta))*diff(phi, zeta) w*(z)=Divide[\[zeta]- 1,a*\[Phi]*(\[zeta])]*D[\[Phi], \[zeta]] Failure Failure
Fail
0.+3.999999998*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
3.999999998+0.*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
0.-3.999999998*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-3.999999998+0.*I <- {w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.10.E31 ϕ ⁢ ( ζ ) = C 1 ⁢ F ⁡ ( b , - a ; b + c ; ζ ) + C 2 ⁢ ζ - b + 1 - c ⁢ F ⁡ ( - a - b - c + 1 , - c + 1 ; 2 - b - c ; ζ ) italic-ϕ 𝜁 subscript 𝐶 1 Gauss-hypergeometric-F 𝑏 𝑎 𝑏 𝑐 𝜁 subscript 𝐶 2 superscript 𝜁 𝑏 1 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑐 1 2 𝑏 𝑐 𝜁 {\displaystyle{\displaystyle\phi(\zeta)=C_{1}F\left(b,-a;b+c;\zeta\right)+C_{2% }\zeta^{-b+1-c}\*F\left(-a-b-c+1,-c+1;2-b-c;\zeta\right)}} phi*(zeta)= C[1]*hypergeom([b, - a], [b + c], zeta)+ C[2]*(zeta)^(- b + 1 - c)* hypergeom([- a - b - c + 1, - c + 1], [2 - b - c], zeta) \[Phi]*(\[zeta])= Subscript[C, 1]*Hypergeometric2F1[b, - a, b + c, \[zeta]]+ Subscript[C, 2]*(\[zeta])^(- b + 1 - c)* Hypergeometric2F1[- a - b - c + 1, - c + 1, 2 - b - c, \[zeta]] Failure Failure Skip Error
32.10.E32 u = ∫ 0 Λ d t t ⁢ ( t - 1 ) ⁢ ( t - z ) 𝑢 superscript subscript 0 Λ 𝑡 𝑡 𝑡 1 𝑡 𝑧 {\displaystyle{\displaystyle u=\int_{0}^{\Lambda}\frac{\mathrm{d}t}{\sqrt{t(t-% 1)(t-z)}}}} u = int((1)/(sqrt(t*(t - 1)*(t - z))), t = 0..Lambda) u = Integrate[Divide[1,Sqrt[t*(t - 1)*(t - z)]], {t, 0, \[CapitalLambda]}] Failure Failure Skip Error
32.10.E33 z ⁢ ( 1 - z ) ⁢ d 2 ϕ d z 2 + ( 1 - 2 ⁢ z ) ⁢ d ϕ d z - 1 4 ⁢ ϕ = 0 𝑧 1 𝑧 derivative italic-ϕ 𝑧 2 1 2 𝑧 derivative italic-ϕ 𝑧 1 4 italic-ϕ 0 {\displaystyle{\displaystyle z(1-z)\frac{{\mathrm{d}}^{2}\phi}{{\mathrm{d}z}^{% 2}}+(1-2z)\frac{\mathrm{d}\phi}{\mathrm{d}z}-\tfrac{1}{4}\phi=0}} z*(1 - z)* diff(phi, [z$(2)])+(1 - 2*z)* diff(phi, z)-(1)/(4)*phi = 0 z*(1 - z)* D[\[Phi], {z, 2}]+(1 - 2*z)* D[\[Phi], z]-Divide[1,4]*\[Phi]= 0 Failure Failure
Fail
-.3535533905-.3535533905*I <- {phi = 2^(1/2)+I*2^(1/2)}
-.3535533905+.3535533905*I <- {phi = 2^(1/2)-I*2^(1/2)}
.3535533905+.3535533905*I <- {phi = -2^(1/2)-I*2^(1/2)}
.3535533905-.3535533905*I <- {phi = -2^(1/2)+I*2^(1/2)}
 Error
32.11.E2 ϕ ⁢ ( x ) = ( 24 ) 1 / 4 ⁢ ( 4 5 ⁢ | x | 5 / 4 - 5 8 ⁢ d 2 ⁢ ln ⁡ | x | ) italic-ϕ 𝑥 superscript 24 1 4 4 5 superscript 𝑥 5 4 5 8 superscript 𝑑 2 𝑥 {\displaystyle{\displaystyle\phi(x)=(24)^{1/4}\left(\tfrac{4}{5}|x|^{5/4}-% \tfrac{5}{8}d^{2}\ln|x|\right)}} phi*(x)=(24)^(1/ 4)*((4)/(5)*(abs(x))^(5/ 4)-(5)/(8)*(d)^(2)* ln(abs(x))) \[Phi]*(x)=(24)^(1/ 4)*(Divide[4,5]*(Abs[x])^(5/ 4)-Divide[5,8]*(d)^(2)* Log[Abs[x]]) Failure Failure
Fail
-.356477509+1.414213562*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 1}
-1.383009716+6.663894383*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 2}
-2.748440825+10.32171246*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 3}
-.356477509-1.414213562*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Error
32.11.E7 ϕ ⁢ ( x ) = 2 3 ⁢ | x | 3 / 2 - 3 4 ⁢ d 2 ⁢ ln ⁡ | x | italic-ϕ 𝑥 2 3 superscript 𝑥 3 2 3 4 superscript 𝑑 2 𝑥 {\displaystyle{\displaystyle\phi(x)=\tfrac{2}{3}|x|^{3/2}-\tfrac{3}{4}d^{2}\ln% |x|}} phi*(x)=(2)/(3)*(abs(x))^(3/ 2)-(3)/(4)*(d)^(2)* ln(abs(x)) \[Phi]*(x)=Divide[2,3]*(Abs[x])^(3/ 2)-Divide[3,4]*(d)^(2)* Log[Abs[x]] Failure Failure
Fail
.7475468953+1.414213562*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 1}
.9428090418+4.907868665*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 2}
.778539070+7.538477552*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 3}
.7475468953-1.414213562*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Error
32.11.E8 d 2 = - π - 1 ⁢ ln ⁡ ( 1 - k 2 ) superscript 𝑑 2 superscript 𝜋 1 1 superscript 𝑘 2 {\displaystyle{\displaystyle d^{2}=-\pi^{-1}\ln\left(1-k^{2}\right)}} (d)^(2)= - (Pi)^(- 1)* ln(1 - (k)^(2)) (d)^(2)= - (Pi)^(- 1)* Log[1 - (k)^(2)] Failure Failure
Fail
Float(-infinity)+3.999999998*I <- {d = 2^(1/2)+I*2^(1/2), k = 1}
.3496991526+4.999999998*I <- {d = 2^(1/2)+I*2^(1/2), k = 2}
.6619068004+4.999999998*I <- {d = 2^(1/2)+I*2^(1/2), k = 3}
Float(-infinity)-3.999999998*I <- {d = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Error
32.11.E9 θ 0 = 3 2 ⁢ d 2 ⁢ ln ⁡ 2 + ph ⁡ Γ ⁡ ( 1 - 1 2 ⁢ i ⁢ d 2 ) + 1 4 ⁢ π ⁢ ( 1 - 2 ⁢ sign ⁡ ( k ) ) subscript 𝜃 0 3 2 superscript 𝑑 2 2 phase Euler-Gamma 1 1 2 𝑖 superscript 𝑑 2 1 4 𝜋 1 2 sign 𝑘 {\displaystyle{\displaystyle\theta_{0}=\tfrac{3}{2}d^{2}\ln 2+\operatorname{ph% }\Gamma\left(1-\tfrac{1}{2}id^{2}\right)+\tfrac{1}{4}\pi(1-2\operatorname{sign% }\left(k\right))}} theta[0]=(3)/(2)*(d)^(2)* ln(2)+ argument(GAMMA(1 -(1)/(2)*I*(d)^(2)))+(1)/(4)*Pi*(1 - 2*signum(k)) Subscript[\[Theta], 0]=Divide[3,2]*(d)^(2)* Log[2]+ Arg[Gamma[1 -Divide[1,2]*I*(d)^(2)]]+Divide[1,4]*Pi*(1 - 2*Sign[k]) Failure Failure
Fail
2.199611726-2.744669520*I <- {d = 2^(1/2)+I*2^(1/2), theta[0] = 2^(1/2)+I*2^(1/2), k = 1}
2.199611726-2.744669520*I <- {d = 2^(1/2)+I*2^(1/2), theta[0] = 2^(1/2)+I*2^(1/2), k = 2}
2.199611726-2.744669520*I <- {d = 2^(1/2)+I*2^(1/2), theta[0] = 2^(1/2)+I*2^(1/2), k = 3}
2.199611726-5.573096644*I <- {d = 2^(1/2)+I*2^(1/2), theta[0] = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Error
32.11.E14 ϕ ⁢ ( x ) = 2 3 ⁢ | x | 3 / 2 + 3 4 ⁢ d 2 ⁢ ln ⁡ | x | italic-ϕ 𝑥 2 3 superscript 𝑥 3 2 3 4 superscript 𝑑 2 𝑥 {\displaystyle{\displaystyle\phi(x)=\tfrac{2}{3}|x|^{3/2}+\tfrac{3}{4}d^{2}\ln% |x|}} phi*(x)=(2)/(3)*(abs(x))^(3/ 2)+(3)/(4)*(d)^(2)* ln(abs(x)) \[Phi]*(x)=Divide[2,3]*(Abs[x])^(3/ 2)+Divide[3,4]*(d)^(2)* Log[Abs[x]] Failure Failure
Fail
.7475468953+1.414213562*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 1}
.9428090418+.748985583*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 2}
.778539070+.946803820*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 3}
.7475468953-1.414213562*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Error
32.11.E15 χ + 3 2 ⁢ d 2 ⁢ ln ⁡ 2 - 1 4 ⁢ π - ph ⁡ Γ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) = n ⁢ π 𝜒 3 2 superscript 𝑑 2 2 1 4 𝜋 phase Euler-Gamma 1 2 𝑖 superscript 𝑑 2 𝑛 𝜋 {\displaystyle{\displaystyle\chi+\tfrac{3}{2}d^{2}\ln 2-\tfrac{1}{4}\pi-% \operatorname{ph}\Gamma\left(\tfrac{1}{2}id^{2}\right)=n\pi}} chi +(3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi - argument(GAMMA((1)/(2)*I*(d)^(2)))= n*Pi \[Chi]+Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi - Arg[Gamma[Divide[1,2]*I*(d)^(2)]]= n*Pi Failure Failure
Fail
-4.083573582+5.573096644*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)+I*2^(1/2), n = 1}
-7.225166236+5.573096644*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)+I*2^(1/2), n = 2}
-10.36675889+5.573096644*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)+I*2^(1/2), n = 3}
-2.512777256-2.744669520*I <- {chi = 2^(1/2)+I*2^(1/2), d = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
 Error
32.11.E17 d 2 = π - 1 ⁢ ln ⁡ ( 1 + k 2 ) superscript 𝑑 2 superscript 𝜋 1 1 superscript 𝑘 2 {\displaystyle{\displaystyle d^{2}=\pi^{-1}\ln\left(1+k^{2}\right)}} (d)^(2)= (Pi)^(- 1)* ln(1 + (k)^(2)) (d)^(2)= (Pi)^(- 1)* Log[1 + (k)^(2)] Failure Failure Skip Error
32.11.E18 χ + 3 2 d 2 ln ⁡ 2 - 1 4 π - ph ⁡ Γ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) < > n π fragments χ 3 2 superscript 𝑑 2 2 1 4 π phase Euler-Gamma 1 2 𝑖 superscript 𝑑 2 n π {\displaystyle{\displaystyle\chi+\tfrac{3}{2}d^{2}\ln 2-\tfrac{1}{4}\pi-% \operatorname{ph}\Gamma\left(\tfrac{1}{2}id^{2}\right)<>n\pi}} chi +(3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi - argument(GAMMA((1)/(2)*I*(d)^(2)))< > n*Pi \[Chi]+Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi - Arg[Gamma[Divide[1,2]*I*(d)^(2)]]< > n*Pi Failure Failure Error Error
32.11.E20 ψ ⁢ ( x ) = 2 3 ⁢ 2 ⁢ x 3 / 2 - 3 2 ⁢ ρ 2 ⁢ ln ⁡ x 𝜓 𝑥 2 3 2 superscript 𝑥 3 2 3 2 superscript 𝜌 2 𝑥 {\displaystyle{\displaystyle\psi(x)=\tfrac{2}{3}\sqrt{2}x^{3/2}-\tfrac{3}{2}% \rho^{2}\ln x}} psi*(x)=(2)/(3)*sqrt(2)*(x)^(3/ 2)-(3)/(2)*(rho)^(2)* ln(x) \[Psi]*(x)=Divide[2,3]*Sqrt[2]*(x)^(3/ 2)-Divide[3,2]*(\[Rho])^(2)* Log[x] Failure Failure
Fail
.4714045206+1.414213562*I <- {psi = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), x = 1}
.1617604583+6.987310206*I <- {psi = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), x = 2}
-.656338799+10.83431442*I <- {psi = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), x = 3}
.4714045206+1.414213562*I <- {psi = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Error
32.11.E21 σ = - sign ⁡ ( ℑ ⁡ s ) 𝜎 sign 𝑠 {\displaystyle{\displaystyle\sigma=-\operatorname{sign}\left(\Im s\right)}} sigma = - signum(Im(s)) \[Sigma]= - Sign[Im[s]] Failure Failure
Fail
2.414213562+1.414213562*I <- {s = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2)}
2.414213562-1.414213562*I <- {s = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)-I*2^(1/2)}
-.414213562-1.414213562*I <- {s = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)-I*2^(1/2)}
-.414213562+1.414213562*I <- {s = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.11.E22 ρ 2 = π - 1 ⁢ ln ⁡ ( ( 1 + | s | 2 ) / | 2 ⁢ ℑ ⁡ s | ) superscript 𝜌 2 superscript 𝜋 1 1 superscript 𝑠 2 2 𝑠 {\displaystyle{\displaystyle\rho^{2}=\pi^{-1}\ln\left((1+|s|^{2})/|2\Im s|% \right)}} (rho)^(2)= (Pi)^(- 1)* ln((1 +(abs(s))^(2))/abs(2*Im(s))) (\[Rho])^(2)= (Pi)^(- 1)* Log[(1 +(Abs[s])^(2))/Abs[2*Im[s]]] Failure Failure
Fail
-.1813465983+3.999999998*I <- {rho = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2)}
-.1813465983+3.999999998*I <- {rho = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2)}
-.1813465983+3.999999998*I <- {rho = 2^(1/2)+I*2^(1/2), s = -2^(1/2)-I*2^(1/2)}
-.1813465983+3.999999998*I <- {rho = 2^(1/2)+I*2^(1/2), s = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.11.E23 θ = - 3 4 ⁢ π - 7 2 ⁢ ρ 2 ⁢ ln ⁡ 2 + ph ⁡ ( 1 + s 2 ) + ph ⁡ Γ ⁡ ( i ⁢ ρ 2 ) 𝜃 3 4 𝜋 7 2 superscript 𝜌 2 2 phase 1 superscript 𝑠 2 phase Euler-Gamma 𝑖 superscript 𝜌 2 {\displaystyle{\displaystyle\theta=-\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\ln{2}% +\operatorname{ph}\left(1+s^{2}\right)+\operatorname{ph}\Gamma\left(i\rho^{2}% \right)}} theta = -(3)/(4)*Pi -(7)/(2)*(rho)^(2)* ln(2)+ argument(1 + (s)^(2))+ argument(GAMMA(I*(rho)^(2))) \[Theta]= -Divide[3,4]*Pi -Divide[7,2]*(\[Rho])^(2)* Log[2]+ Arg[1 + (s)^(2)]+ Arg[Gamma[I*(\[Rho])^(2)]] Failure Failure
Fail
.873794061+11.11827408*I <- {rho = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)}
.873794061+8.289846961*I <- {rho = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)}
-1.954633063+8.289846961*I <- {rho = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)}
-1.954633063+11.11827408*I <- {rho = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.11.E27 σ = ( 2 / π ) ⁢ arcsin ⁡ ( π ⁢ λ ) 𝜎 2 𝜋 𝜋 𝜆 {\displaystyle{\displaystyle\sigma=(2/\pi)\operatorname{arcsin}\left(\pi% \lambda\right)}} sigma =(2/ Pi)* arcsin(Pi*lambda) \[Sigma]=(2/ Pi)* ArcSin[Pi*\[Lambda]] Failure Failure
Fail
.9182444652-.197124803*I <- {lambda = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2)}
.9182444652-3.025551927*I <- {lambda = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)-I*2^(1/2)}
-1.910182659-3.025551927*I <- {lambda = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)-I*2^(1/2)}
-1.910182659-.197124803*I <- {lambda = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.11.E28 B = 2 - 2 ⁢ σ ⁢ Γ 2 ⁡ ( 1 2 ⁢ ( 1 - σ ) ) ⁢ Γ ⁡ ( 1 2 ⁢ ( 1 + σ ) + ν ) Γ 2 ⁡ ( 1 2 ⁢ ( 1 + σ ) ) ⁢ Γ ⁡ ( 1 2 ⁢ ( 1 - σ ) + ν ) 𝐵 superscript 2 2 𝜎 Euler-Gamma 2 1 2 1 𝜎 Euler-Gamma 1 2 1 𝜎 𝜈 Euler-Gamma 2 1 2 1 𝜎 Euler-Gamma 1 2 1 𝜎 𝜈 {\displaystyle{\displaystyle B=2^{-2\sigma}\frac{{\Gamma^{2}}\left(\tfrac{1}{2% }(1-\sigma)\right)\Gamma\left(\tfrac{1}{2}(1+\sigma)+\nu\right)}{{\Gamma^{2}}% \left(\tfrac{1}{2}(1+\sigma)\right)\Gamma\left(\tfrac{1}{2}(1-\sigma)+\nu% \right)}}} B = (2)^(- 2*sigma)*((GAMMA((1)/(2)*(1 - sigma)))^(2)* GAMMA((1)/(2)*(1 + sigma)+ nu))/((GAMMA((1)/(2)*(1 + sigma)))^(2)* GAMMA((1)/(2)*(1 - sigma)+ nu)) B = (2)^(- 2*\[Sigma])*Divide[(Gamma[Divide[1,2]*(1 - \[Sigma])])^(2)* Gamma[Divide[1,2]*(1 + \[Sigma])+ \[Nu]],(Gamma[Divide[1,2]*(1 + \[Sigma])])^(2)* Gamma[Divide[1,2]*(1 - \[Sigma])+ \[Nu]]] Failure Failure
Fail
1.341555906+1.650216622*I <- {B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2)}
3.846316726+1.915270556*I <- {B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)-I*2^(1/2)}
.222646953-2.456175086*I <- {B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)-I*2^(1/2)}
1.808639547+1.332954715*I <- {B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.11.E31 h * = 1 / ( π 1 / 2 ⁢ Γ ⁡ ( ν + 1 ) ) superscript ℎ 1 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 {\displaystyle{\displaystyle h^{*}=\ifrac{1}{\left(\pi^{1/2}\Gamma\left(\nu+1% \right)\right)}}} (h)^(*)=(1)/((Pi)^(1/ 2)* GAMMA(nu + 1)) (h)^(*)=Divide[1,(Pi)^(1/ 2)* Gamma[\[Nu]+ 1]] Error Failure - Error
32.11.E34 ϕ ⁢ ( x ) = 1 3 ⁢ 3 ⁢ x 2 - 4 3 ⁢ d 2 ⁢ 3 ⁢ ln ⁡ ( 2 ⁢ | x | ) italic-ϕ 𝑥 1 3 3 superscript 𝑥 2 4 3 superscript 𝑑 2 3 2 𝑥 {\displaystyle{\displaystyle\phi(x)=\tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2% }\sqrt{3}\ln\left(\sqrt{2}|x|\right)}} phi*(x)=(1)/(3)*sqrt(3)*(x)^(2)-(4)/(3)*(d)^(2)*sqrt(3)*ln(sqrt(2)*abs(x)) \[Phi]*(x)=Divide[1,3]*Sqrt[3]*(x)^(2)-Divide[4,3]*(d)^(2)*Sqrt[3]*Log[Sqrt[2]*Abs[x]] Failure Failure
Fail
.8368632927+4.615723248*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 1}
.519026047+12.43295619*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 2}
-.953511738+17.59269598*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 3}
.8368632927+1.787296124*I <- {d = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Error
32.11.E35 d 2 = - 1 4 ⁢ 3 ⁢ π - 1 ⁢ ln ⁡ ( 1 - | μ | 2 ) superscript 𝑑 2 1 4 3 superscript 𝜋 1 1 superscript 𝜇 2 {\displaystyle{\displaystyle d^{2}=-\tfrac{1}{4}\sqrt{3}\pi^{-1}\ln\left(1-|% \mu|^{2}\right)}} (d)^(2)= -(1)/(4)*sqrt(3)*(Pi)^(- 1)* ln(1 -(abs(mu))^(2)) (d)^(2)= -Divide[1,4]*Sqrt[3]*(Pi)^(- 1)* Log[1 -(Abs[\[Mu]])^(2)] Failure Failure
Fail
.1514241750+4.433012700*I <- {d = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}
.1514241750+4.433012700*I <- {d = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}
.1514241750+4.433012700*I <- {d = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}
.1514241750+4.433012700*I <- {d = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.11.E36 θ 0 = 1 3 ⁢ d 2 ⁢ 3 ⁢ ln ⁡ 3 + 2 3 ⁢ π ⁢ ν + 7 12 ⁢ π + ph ⁡ μ + ph ⁡ Γ ⁡ ( - 2 3 ⁢ i ⁢ 3 ⁢ d 2 ) subscript 𝜃 0 1 3 superscript 𝑑 2 3 3 2 3 𝜋 𝜈 7 12 𝜋 phase 𝜇 phase Euler-Gamma 2 3 𝑖 3 superscript 𝑑 2 {\displaystyle{\displaystyle\theta_{0}=\tfrac{1}{3}d^{2}\sqrt{3}\ln 3+\tfrac{2% }{3}\pi\nu+\tfrac{7}{12}\pi+\operatorname{ph}\mu+\operatorname{ph}\Gamma\left(% -\tfrac{2}{3}i\sqrt{3}d^{2}\right)}} theta[0]=(1)/(3)*(d)^(2)*sqrt(3)*ln(3)+(2)/(3)*Pi*nu +(7)/(12)*Pi + argument(mu)+ argument(GAMMA(-(2)/(3)*I*sqrt(3)*(d)^(2))) Subscript[\[Theta], 0]=Divide[1,3]*(d)^(2)*Sqrt[3]*Log[3]+Divide[2,3]*Pi*\[Nu]+Divide[7,12]*Pi + Arg[\[Mu]]+ Arg[Gamma[-Divide[2,3]*I*Sqrt[3]*(d)^(2)]] Failure Failure
Fail
-4.165702275-4.084844799*I <- {d = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta[0] = 2^(1/2)+I*2^(1/2)}
-4.165702275-6.913271923*I <- {d = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta[0] = 2^(1/2)-I*2^(1/2)}
-6.994129399-6.913271923*I <- {d = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta[0] = -2^(1/2)-I*2^(1/2)}
-6.994129399-4.084844799*I <- {d = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta[0] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.11.E37 μ = 1 + ( 2 ⁢ i ⁢ h ⁢ π 3 / 2 ⁢ exp ⁡ ( - i ⁢ π ⁢ ν ) / Γ ⁡ ( - ν ) ) 𝜇 1 2 𝑖 ℎ superscript 𝜋 3 2 𝑖 𝜋 𝜈 Euler-Gamma 𝜈 {\displaystyle{\displaystyle\mu=1+\left(\ifrac{2ih\pi^{3/2}\exp\left(-i\pi\nu% \right)}{\Gamma\left(-\nu\right)}\right)}} mu = 1 +((2*I*h*(Pi)^(3/ 2)* exp(- I*Pi*nu))/(GAMMA(- nu))) \[Mu]= 1 +(Divide[2*I*h*(Pi)^(3/ 2)* Exp[- I*Pi*\[Nu]],Gamma[- \[Nu]]]) Failure Failure
Fail
-14237.77906+14002.23390*I <- {h = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2)}
3.087580692+.7179287085*I <- {h = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2)}
-.2354286281+1.240678506*I <- {h = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2)}
-3486.428322-3384.777319*I <- {h = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.13.E6 u x ⁢ t = sin ⁡ u subscript 𝑢 𝑥 𝑡 𝑢 {\displaystyle{\displaystyle u_{xt}=\sin u}} u[x*t]= sin(u) Subscript[u, x*t]= Sin[u] Failure Failure
Fail
-.737321978+1.112452092*I <- {u = 2^(1/2)+I*2^(1/2), u[x*t] = 2^(1/2)+I*2^(1/2)}
-.737321978-1.715975032*I <- {u = 2^(1/2)+I*2^(1/2), u[x*t] = 2^(1/2)-I*2^(1/2)}
-3.565749102-1.715975032*I <- {u = 2^(1/2)+I*2^(1/2), u[x*t] = -2^(1/2)-I*2^(1/2)}
-3.565749102+1.112452092*I <- {u = 2^(1/2)+I*2^(1/2), u[x*t] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
32.14.E2 F ⁢ ( s ) = exp ⁡ ( - ∫ s ∞ ( x - s ) ⁢ w 2 ⁢ ( x ) ⁢ d x ) 𝐹 𝑠 superscript subscript 𝑠 𝑥 𝑠 superscript 𝑤 2 𝑥 𝑥 {\displaystyle{\displaystyle F(s)=\exp\left(-\int_{s}^{\infty}(x-s)w^{2}(x)% \mathrm{d}x\right)}} F*(s)= exp(- int((x - s)* (w)^(2)*(x), x = s..infinity)) F*(s)= Exp[- Integrate[(x - s)* (w)^(2)*(x), {x, s, Infinity}]] Failure Failure Skip Error
32.15.E1 ∫ - ∞ ∞ exp ⁡ ( - 1 4 ⁢ ξ 4 - z ⁢ ξ 2 ) ⁢ p m ⁢ ( ξ ) ⁢ p n ⁢ ( ξ ) ⁢ d ξ = δ m , n superscript subscript 1 4 superscript 𝜉 4 𝑧 superscript 𝜉 2 subscript 𝑝 𝑚 𝜉 subscript 𝑝 𝑛 𝜉 𝜉 Kronecker 𝑚 𝑛 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}\exp\left(-\tfrac{1}{4}\xi^% {4}-z\xi^{2}\right)p_{m}(\xi)p_{n}(\xi)\mathrm{d}\xi=\delta_{m,n}}} int(exp(-(1)/(4)*(xi)^(4)- z*(xi)^(2))*p[m]*(xi)* p[n]*(xi), xi = - infinity..infinity)= KroneckerDelta[m, n] Integrate[Exp[-Divide[1,4]*(\[Xi])^(4)- z*(\[Xi])^(2)]*Subscript[p, m]*(\[Xi])* Subscript[p, n]*(\[Xi]), {\[Xi], - Infinity, Infinity}]= KroneckerDelta[m, n] Failure Failure Skip Error
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