# Results of Heun Functions

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DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
31.2.E1 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d% }w}{\mathrm{d}z}+\frac{\alpha\beta z-q}{z(z-1)(z-a)}w=0}}$ diff(w, [z$(2)])+((gamma)/(z)+(delta)/(z - 1)+(epsilon)/(z - a))* diff(w, z)+(alpha*beta*z - q)/(z*(z - 1)*(z - a))*w = 0 D[w, {z, 2}]+(Divide[\[Gamma],z]+Divide[\[Delta],z - 1]+Divide[\[Epsilon],z - a])* D[w, z]+Divide[\[Alpha]*\[Beta]*z - q,z*(z - 1)*(z - a)]*w = 0 Failure Failure Skip Successful 31.2.E5 ${\displaystyle{\displaystyle z={\sin^{2}}\theta}}$ z = (sin(theta))^(2) z = (Sin[\[Theta]])^(2) Failure Failure Fail -3.123831633+.115712505*I <- {theta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -3.123831633-2.712714619*I <- {theta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -5.952258757-2.712714619*I <- {theta = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -5.952258757+.115712505*I <- {theta = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data Fail Complex[-3.1238316385762896, 0.11571250751855189] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-3.1238316385762896, 2.7127146172276384] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-3.1238316385762896, 0.11571250751855189] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-3.1238316385762896, 2.7127146172276384] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data 31.2.E6 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\theta}^{2}}+% \left({(2\gamma-1)\cot\theta-(2\delta-1)\tan\theta}-\frac{\epsilon\sin\left(2% \theta\right)}{a-{\sin^{2}}\theta}\right)\frac{\mathrm{d}w}{\mathrm{d}\theta}+% 4\frac{\alpha\beta{\sin^{2}}\theta-q}{a-{\sin^{2}}\theta}w=0}}$ diff(w, [theta$(2)])+((2*gamma - 1)* cot(theta)-(2*delta - 1)* tan(theta)-(epsilon*sin(2*theta))/(a - (sin(theta))^(2)))* diff(w, theta)+ 4*(alpha*beta*(sin(theta))^(2)- q)/(a - (sin(theta))^(2))*w = 0 D[w, {\[Theta], 2}]+((2*\[Gamma]- 1)* Cot[\[Theta]]-(2*\[Delta]- 1)* Tan[\[Theta]]-Divide[\[Epsilon]*Sin[2*\[Theta]],a - (Sin[\[Theta]])^(2)])* D[w, \[Theta]]+ 4*Divide[\[Alpha]*\[Beta]*(Sin[\[Theta]])^(2)- q,a - (Sin[\[Theta]])^(2)]*w = 0 Failure Failure Skip Skip
31.2#Ex8 ${\displaystyle{\displaystyle z={\operatorname{sn}^{2}}\left(\zeta,k\right)}}$ z = (JacobiSN(zeta, k))^(2) z = (JacobiSN[\[zeta], (k)^2])^(2) Failure Failure
Fail
.165123316+1.322794733*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1}
1.289691928+.142135589*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 2}
1.298236003+1.527150376*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 3}
.165123316+1.505632391*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
Error
31.2.E8 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+% \left((2\gamma-1)\frac{\operatorname{cn}\zeta\operatorname{dn}\zeta}{% \operatorname{sn}\zeta}-(2\delta-1)\frac{\operatorname{sn}\zeta\operatorname{% dn}\zeta}{\operatorname{cn}\zeta}-(2\epsilon-1)k^{2}\frac{\operatorname{sn}% \zeta\operatorname{cn}\zeta}{\operatorname{dn}\zeta}\right)\frac{\mathrm{d}w}{% \mathrm{d}\zeta}+4k^{2}(\alpha\beta{\operatorname{sn}^{2}}\zeta-q)w=0}}$ diff(w, [zeta$(2)])+((2*gamma - 1)*(JacobiCN(zeta, k)*JacobiDN(zeta, k))/(JacobiSN(zeta, k))-(2*delta - 1)*(JacobiSN(zeta, k)*JacobiDN(zeta, k))/(JacobiCN(zeta, k))-(2*epsilon - 1)*(k)^(2)*(JacobiSN(zeta, k)*JacobiCN(zeta, k))/(JacobiDN(zeta, k)))* diff(w, zeta)+ 4*(k)^(2)*(alpha*beta*(JacobiSN(zeta, k))^(2)- q)* w = 0 D[w, {\[zeta], 2}]+((2*\[Gamma]- 1)*Divide[JacobiCN[\[zeta], (k)^2]*JacobiDN[\[zeta], (k)^2],JacobiSN[\[zeta], (k)^2]]-(2*\[Delta]- 1)*Divide[JacobiSN[\[zeta], (k)^2]*JacobiDN[\[zeta], (k)^2],JacobiCN[\[zeta], (k)^2]]-(2*\[Epsilon]- 1)*(k)^(2)*Divide[JacobiSN[\[zeta], (k)^2]*JacobiCN[\[zeta], (k)^2],JacobiDN[\[zeta], (k)^2]])* D[w, \[zeta]]+ 4*(k)^(2)*(\[Alpha]*\[Beta]*(JacobiSN[\[zeta], (k)^2])^(2)- q)* w = 0 Failure Failure Skip Error 31.3.E1 ${\displaystyle{\displaystyle\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,% \delta;z\right)=\sum_{j=0}^{\infty}c_{j}z^{j}}}$ HeunG(a, q, alpha, beta, gamma, delta, z)= sum(c[j]*(z)^(j), j = 0..infinity) Error Failure Error Skip - 31.7.E1 ${\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=% \mathit{H\!\ell}\left(1,\alpha\beta;\alpha,\beta,\gamma,\delta;z\right)}}$ hypergeom([alpha , beta], [gamma], z)= HeunG(1, alpha*beta, alpha, beta, gamma, delta, z) Error Successful Error - - 31.7.E1 ${\displaystyle{\displaystyle\mathit{H\!\ell}\left(1,\alpha\beta;\alpha,\beta,% \gamma,\delta;z\right)=\mathit{H\!\ell}\left(0,0;\alpha,\beta,\gamma,\alpha+% \beta+1-\gamma;z\right)}}$ HeunG(1, alpha*beta, alpha, beta, gamma, delta, z)= HeunG(0, 0, alpha, beta, gamma, alpha + beta + 1 - gamma, z) Error Successful Error - - 31.7.E1 ${\displaystyle{\displaystyle\mathit{H\!\ell}\left(0,0;\alpha,\beta,\gamma,% \alpha+\beta+1-\gamma;z\right)=\mathit{H\!\ell}\left(a,a\alpha\beta;\alpha,% \beta,\gamma,\alpha+\beta+1-\gamma;z\right)}}$ HeunG(0, 0, alpha, beta, gamma, alpha + beta + 1 - gamma, z)= HeunG(a, a*alpha*beta, alpha, beta, gamma, alpha + beta + 1 - gamma, z) Error Successful Error - - 31.7.E2 ${\displaystyle{\displaystyle\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,% \gamma,\alpha+\beta-2\gamma+1;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,% \tfrac{1}{2}\beta;\gamma;1-(1-z)^{2}\right)}}$ HeunG(2, alpha*beta, alpha, beta, gamma, alpha + beta - 2*gamma + 1, z)= hypergeom([(1)/(2)*alpha ,(1)/(2)*beta], [gamma], 1 -(1 - z)^(2)) Error Failure Error Skip - 31.7.E3 ${\displaystyle{\displaystyle\mathit{H\!\ell}\left(4,\alpha\beta;\alpha,\beta,% \tfrac{1}{2},\tfrac{2}{3}(\alpha+\beta);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{% 3}\alpha,\tfrac{1}{3}\beta;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1}{4}z)\right)}}$ HeunG(4, alpha*beta, alpha, beta, (1)/(2), (2)/(3)*(alpha + beta), z)= hypergeom([(1)/(3)*alpha ,(1)/(3)*beta], [(1)/(2)], 1 -(1 - z)^(2)*(1 -(1)/(4)*z)) Error Failure Error Fail -3.621930481+3.545214886*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.7797065831-28.11306461*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -2.255705377+1.783715852*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -2.255705377-1.783715852*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} ... skip entries to safe data - 31.7.E4 ${\displaystyle{\displaystyle\mathit{H\!\ell}\left(\tfrac{1}{2}+i\tfrac{\sqrt{3% }}{2},\alpha\beta(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}% (\alpha+\beta+1),\tfrac{1}{3}(\alpha+\beta+1);z\right)={{}_{2}F_{1}}\left(% \tfrac{1}{3}\alpha,\tfrac{1}{3}\beta;\tfrac{1}{3}(\alpha+\beta+1);1-\left(1-% \left(\tfrac{3}{2}-i\tfrac{\sqrt{3}}{2}\right)z\right)^{3}\right)}}$ HeunG((1)/(2)+ I*(sqrt(3))/(2), alpha*beta*((1)/(2)+ I*(sqrt(3))/(6)), alpha, beta, (1)/(3)*(alpha + beta + 1), (1)/(3)*(alpha + beta + 1), z)= hypergeom([(1)/(3)*alpha ,(1)/(3)*beta], [(1)/(3)*(alpha + beta + 1)], 1 -(1 -((3)/(2)- I*(sqrt(3))/(2))*z)^(3)) Error Failure Error Fail 2.753353716-2.379263249*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -2.603938147-2.727318150*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} .8968927627+1.268651452*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} .3609263936-1.161063097*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} ... skip entries to safe data - 31.9.E2 ${\displaystyle{\displaystyle\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{% \delta-1}(t-a)^{\epsilon-1}\*w_{m}(t)w_{k}(t)\mathrm{d}t=\delta_{m,k}\theta_{m% }}}$ int((t)^(gamma - 1)*(1 - t)^(delta - 1)*(t - a)^(epsilon - 1)* w[m]*(t)* w[k]*(t), t = zeta..(1 + , 0 + , 1 - , 0 -))= KroneckerDelta[m, k]*theta[m] Integrate[(t)^(\[Gamma]- 1)*(1 - t)^(\[Delta]- 1)*(t - a)^(\[Epsilon]- 1)* Subscript[w, m]*(t)* Subscript[w, k]*(t), {t, \[zeta], (1 + , 0 + , 1 - , 0 -)}]= KroneckerDelta[m, k]*Subscript[\[Theta], m] Error Failure - Error 31.9#Ex1 ${\displaystyle{\displaystyle f_{0}(q_{m},z)=\mathit{H\!\ell}\left(a,q_{m};% \alpha,\beta,\gamma,\delta;z\right)}}$ f[0]*(q[m], z)= HeunG(a, q[m], alpha, beta, gamma, delta, z) Error Failure Error Skip - 31.9#Ex2 ${\displaystyle{\displaystyle f_{1}(q_{m},z)=\mathit{H\!\ell}\left(1-a,\alpha% \beta-q_{m};\alpha,\beta,\delta,\gamma;1-z\right)}}$ f[1]*(q[m], z)= HeunG(1 - a, alpha*beta - q[m], alpha, beta, delta, gamma, 1 - z) Error Failure Error Skip - 31.10#Ex1 ${\displaystyle{\displaystyle\cos\theta=\left(\frac{zt}{a}\right)^{1/2}}}$ cos(theta)=((z*t)/(a))^(1/ 2) Cos[\[Theta]]=(Divide[z*t,a])^(1/ 2) Failure Failure Fail -.9668889726-2.452589209*I <- {a = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.9668889726-1.370197009*I <- {a = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -.2015221077-.604830144*I <- {a = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -.2015221077-3.217956074*I <- {a = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data Skip 31.10#Ex2 ${\displaystyle{\displaystyle\sin\theta\cos\phi=\mathrm{i}\left(\frac{(z-a)(t-a% )}{a(1-a)}\right)^{1/2}}}$ sin(theta)*cos(phi)= I*(((z - a)*(t - a))/(a*(1 - a)))^(1/ 2) Sin[\[Theta]]*Cos[\[Phi]]= I*(Divide[(z - a)*(t - a),a*(1 - a)])^(1/ 2) Failure Failure Skip Skip 31.10#Ex3 ${\displaystyle{\displaystyle\sin\theta\sin\phi=\left(\frac{(z-1)(t-1)}{1-a}% \right)^{1/2}}}$ sin(theta)*sin(phi)=(((z - 1)*(t - 1))/(1 - a))^(1/ 2) Sin[\[Theta]]*Sin[\[Phi]]=(Divide[(z - 1)*(t - 1),1 - a])^(1/ 2) Failure Failure Skip Skip 31.10.E8 ${\displaystyle{\displaystyle{\sin^{2}}\theta\left(\frac{{\partial}^{2}\mathcal% {K}}{{\partial\theta}^{2}}+\left((1-2\gamma)\tan\theta+2(\delta+\epsilon-% \tfrac{1}{2})\cot\theta\right)\frac{\partial\mathcal{K}}{\partial\theta}-4% \alpha\beta\mathcal{K}\right)+\frac{{\partial}^{2}\mathcal{K}}{{\partial\phi}^% {2}}+\left((1-2\delta)\cot\phi-(1-2\epsilon)\tan\phi\right)\frac{\partial% \mathcal{K}}{\partial\phi}=0}}$ (sin(theta))^(2)*(diff(K, [theta$(2)])+((1 - 2*gamma)*tan(theta)+ 2*(delta + epsilon -(1)/(2))*cot(theta))*diff(K, theta)- 4*alpha*beta*K)+ diff(K, [phi$(2)])+((1 - 2*delta)*cot(phi)-(1 - 2*epsilon)*tan(phi))* diff(K, phi)= 0 (Sin[\[Theta]])^(2)*(D[K, {\[Theta], 2}]+((1 - 2*\[Gamma])*Tan[\[Theta]]+ 2*(\[Delta]+ \[Epsilon]-Divide[1,2])*Cot[\[Theta]])*D[K, \[Theta]]- 4*\[Alpha]*\[Beta]*K)+ D[K, {\[Phi], 2}]+((1 - 2*\[Delta])*Cot[\[Phi]]-(1 - 2*\[Epsilon])*Tan[\[Phi]])* D[K, \[Phi]]= 0 Failure Failure Fail 132.0659658-73.30251604*I <- {K = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} 73.30251604-132.0659658*I <- {K = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} 132.0659658-73.30251604*I <- {K = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 73.30251604-132.0659658*I <- {K = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data Fail Complex[132.06596595801318, -73.30251627701908] <- {Rule[K, 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]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[73.30251627701908, -132.06596595801318] <- {Rule[K, 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]]]], Rule[θ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[132.06596595801318, -73.30251627701908] <- {Rule[K, 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]]]], Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[73.30251627701908, -132.06596595801318] <- {Rule[K, 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]]]], Rule[θ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data 31.10.E10 ${\displaystyle{\displaystyle\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma% }\*{{}_{2}F_{1}}\left({\frac{1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-% \sigma+\beta\atop\gamma};\frac{zt}{a}\right)\*{{}_{2}F_{1}}\left({-\frac{1}{2}% +\delta+\sigma,-\frac{1}{2}+\epsilon-\sigma\atop\delta};\frac{a(z-1)(t-1)}{(a-% 1)(zt-a)}\right)}}$ K*(z , t)=(z*t - a)^((1)/(2)- delta - sigma)* hypergeom([(1)/(2)- delta - sigma + alpha ,(1)/(2)- delta - sigma + beta], [gamma], (z*t)/(a))* hypergeom([-(1)/(2)+ delta + sigma , -(1)/(2)+ epsilon - sigma], [delta], (a*(z - 1)*(t - 1))/((a - 1)*(z*t - a))) K*(z , t)=(z*t - a)^(Divide[1,2]- \[Delta]- \[Sigma])* HypergeometricPFQ[{Divide[1,2]- \[Delta]- \[Sigma]+ \[Alpha],Divide[1,2]- \[Delta]- \[Sigma]+ \[Beta]}, {\[Gamma]}, Divide[z*t,a]]* HypergeometricPFQ[{-Divide[1,2]+ \[Delta]+ \[Sigma], -Divide[1,2]+ \[Epsilon]- \[Sigma]}, {\[Delta]}, Divide[a*(z - 1)*(t - 1),(a - 1)*(z*t - a)]] Failure Failure Skip Error 31.10.E18 ${\displaystyle{\displaystyle\frac{{\partial}^{2}\mathcal{K}}{{\partial u}^{2}}% +\frac{{\partial}^{2}\mathcal{K}}{{\partial v}^{2}}+\frac{{\partial}^{2}% \mathcal{K}}{{\partial w}^{2}}+\frac{2\gamma-1}{u}\frac{\partial\mathcal{K}}{% \partial u}+\frac{2\delta-1}{v}\frac{\partial\mathcal{K}}{\partial v}+\frac{2% \epsilon-1}{w}\frac{\partial\mathcal{K}}{\partial w}=0}}$ diff(K, [u$(2)])+ diff(K, [v$(2)])+ diff(K, [w$(2)])+(2*gamma - 1)/(u)*diff(K, u)+(2*delta - 1)/(v)*diff(K, v)+(2*epsilon - 1)/(w)*diff(K, w)= 0 D[K, {u, 2}]+ D[K, {v, 2}]+ D[K, {w, 2}]+Divide[2*\[Gamma]- 1,u]*D[K, u]+Divide[2*\[Delta]- 1,v]*D[K, v]+Divide[2*\[Epsilon]- 1,w]*D[K, w]= 0 Successful Successful - -
31.10#Ex7 ${\displaystyle{\displaystyle u=r\cos\theta}}$ u = r*cos(theta) u = r*Cos[\[Theta]] Failure Failure
Fail
-1.769276062+3.636960052*I <- {r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2)}
-1.769276062+.808532928*I <- {r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), u = 2^(1/2)-I*2^(1/2)}
-4.597703186+.808532928*I <- {r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), u = -2^(1/2)-I*2^(1/2)}
-4.597703186+3.636960052*I <- {r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), u = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-1.7692760628877078, 3.636960055953727] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.636960055953727, -1.7692760628877078] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7692760628877078, 3.636960055953727] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.636960055953727, -1.7692760628877078] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
31.10#Ex8 ${\displaystyle{\displaystyle v=r\sin\theta\sin\phi}}$ v = r*sin(theta)*sin(phi) v = r*Sin[\[Theta]]*Sin[\[Phi]] Failure Failure
Fail
-3.167193693-6.839909303*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2)}
-3.167193693-9.668336427*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2)}
-5.995620817-9.668336427*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), v = -2^(1/2)-I*2^(1/2)}
-5.995620817-6.839909303*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), v = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-3.1671937049405976, -6.839909310002729] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.261108038166927, -5.261108038166927] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[5.9956208296867874, 9.66833643474892] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[8.089535162913117, 8.089535162913117] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
31.10#Ex9 ${\displaystyle{\displaystyle w=r\sin\theta\cos\phi}}$ w = r*sin(theta)*cos(phi) w = r*Sin[\[Theta]]*Cos[\[Phi]] Failure Failure
Fail
-6.105916754+5.235877121*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
-6.105916754+2.407449997*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
-8.934343878+2.407449997*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-8.934343878+5.235877121*I <- {phi = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-6.1059167609067115, 5.235877134071633] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[7.1571861513265675, -4.764438262814106] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-6.1059167609067115, 5.235877134071633] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[7.1571861513265675, -4.764438262814106] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
31.10.E21 ${\displaystyle{\displaystyle\frac{{\partial}^{2}\mathcal{K}}{{\partial r}^{2}}% +\frac{2(\gamma+\delta+\epsilon)-1}{r}\frac{\partial\mathcal{K}}{\partial r}+% \frac{1}{r^{2}}\frac{{\partial}^{2}\mathcal{K}}{{\partial\theta}^{2}}+\frac{(2% (\delta+\epsilon)-1)\cot\theta-(2\gamma-1)\tan\theta}{r^{2}}\frac{\partial% \mathcal{K}}{\partial\theta}+\frac{1}{r^{2}{\sin^{2}}\theta}\frac{{\partial}^{% 2}\mathcal{K}}{{\partial\phi}^{2}}+\frac{(2\delta-1)\cot\phi-(2\epsilon-1)\tan% \phi}{r^{2}{\sin^{2}}\theta}\frac{\partial\mathcal{K}}{\partial\phi}=0}}$ diff(K, [r$(2)])+(2*(gamma + delta + epsilon)- 1)/(r)*diff(K, r)+(1)/((r)^(2))*diff(K, [theta$(2)])+((2*(delta + epsilon)- 1)* cot(theta)-(2*gamma - 1)* tan(theta))/((r)^(2))*diff(K, theta)+(1)/((r)^(2)* (sin(theta))^(2))*diff(K, [phi$(2)])+((2*delta - 1)* cot(phi)-(2*epsilon - 1)* tan(phi))/((r)^(2)* (sin(theta))^(2))*diff(K, phi)= 0 D[K, {r, 2}]+Divide[2*(\[Gamma]+ \[Delta]+ \[Epsilon])- 1,r]*D[K, r]+Divide[1,(r)^(2)]*D[K, {\[Theta], 2}]+Divide[(2*(\[Delta]+ \[Epsilon])- 1)* Cot[\[Theta]]-(2*\[Gamma]- 1)* Tan[\[Theta]],(r)^(2)]*D[K, \[Theta]]+Divide[1,(r)^(2)* (Sin[\[Theta]])^(2)]*D[K, {\[Phi], 2}]+Divide[(2*\[Delta]- 1)* Cot[\[Phi]]-(2*\[Epsilon]- 1)* Tan[\[Phi]],(r)^(2)* (Sin[\[Theta]])^(2)]*D[K, \[Phi]]= 0 Successful Successful - - 31.11.E12 ${\displaystyle{\displaystyle P_{j}=\frac{\Gamma\left(\alpha+j\right)\Gamma% \left(1-\gamma+\alpha+j\right)}{\Gamma\left(1+\alpha-\beta+\epsilon+2j\right)}% z^{-\alpha-j}\*{{}_{2}F_{1}}\left({\alpha+j,1-\gamma+\alpha+j\atop 1+\alpha-% \beta+\epsilon+2j};\frac{1}{z}\right)}}$ P[j]=(GAMMA(alpha + j)*GAMMA(1 - gamma + alpha + j))/(GAMMA(1 + alpha - beta + epsilon + 2*j))*(z)^(- alpha - j)* hypergeom([alpha + j , 1 - gamma + alpha + j], [1 + alpha - beta + epsilon + 2*j], (1)/(z)) Subscript[P, j]=Divide[Gamma[\[Alpha]+ j]*Gamma[1 - \[Gamma]+ \[Alpha]+ j],Gamma[1 + \[Alpha]- \[Beta]+ \[Epsilon]+ 2*j]]*(z)^(- \[Alpha]- j)* HypergeometricPFQ[{\[Alpha]+ j , 1 - \[Gamma]+ \[Alpha]+ j}, {1 + \[Alpha]- \[Beta]+ \[Epsilon]+ 2*j}, Divide[1,z]] Failure Failure Skip Error 31.12.E1 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon\right)\frac{\mathrm{d}w}{\mathrm{% d}z}+\frac{\alpha z-q}{z(z-1)}w=0}}$ diff(w, [z$(2)])+((gamma)/(z)+(delta)/(z - 1)+ epsilon)* diff(w, z)+(alpha*z - q)/(z*(z - 1))*w = 0 D[w, {z, 2}]+(Divide[\[Gamma],z]+Divide[\[Delta],z - 1]+ \[Epsilon])* D[w, z]+Divide[\[Alpha]*z - q,z*(z - 1)]*w = 0 Failure Failure
Fail
1.414213562+1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1.414213562+1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.414213562-1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.414213562-1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 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
Error
31.12.E2 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \frac{\delta}{z^{2}}+\frac{\gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+% \frac{\alpha z-q}{z^{2}}w=0}}$ diff(w, [z$(2)])+((delta)/((z)^(2))+(gamma)/(z)+ 1)* diff(w, z)+(alpha*z - q)/((z)^(2))*w = 0 D[w, {z, 2}]+(Divide[\[Delta],(z)^(2)]+Divide[\[Gamma],z]+ 1)* D[w, z]+Divide[\[Alpha]*z - q,(z)^(2)]*w = 0 Failure Failure Fail .4142135620+1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.4142135620+1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -2.414213562-1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} 2.414213562-1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 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 Error 31.12.E3 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(% \frac{\gamma}{z}+\delta+z\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z% -q}{z}w=0}}$ diff(w, [z$(2)])-((gamma)/(z)+ delta + z)* diff(w, z)+(alpha*z - q)/(z)*w = 0 D[w, {z, 2}]-(Divide[\[Gamma],z]+ \[Delta]+ z)* D[w, z]+Divide[\[Alpha]*z - q,z]*w = 0 Failure Failure
Fail
-1.414213562+2.585786436*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.414213562+2.585786436*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.414213562+5.414213560*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.414213562+5.414213560*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 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
Error
31.12.E4 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \gamma+z\right)z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\alpha z-q\right)w=0}}$ diff(w, [z$(2)])+(gamma + z)* z*diff(w, z)+(alpha*z - q)* w = 0 D[w, {z, 2}]+(\[Gamma]+ z)* z*D[w, z]+(\[Alpha]*z - q)* w = 0 Failure Failure Fail -5.656854245+1.656854247*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} 5.656854245+1.656854247*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 5.656854245-9.656854243*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -5.656854245-9.656854243*I <- {alpha = 2^(1/2)+I*2^(1/2), q = 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 Error 31.15.E1 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \sum_{j=1}^{N}\frac{\gamma_{j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}% +\frac{\Phi(z)}{\prod_{j=1}^{N}(z-a_{j})}w=0}}$ diff(w, [z$(2)])+(sum((gamma[j])/(z - a[j]), j = 1..N))* diff(w, z)+(Phi*(z))/(product(z - a[j], j = 1..N))*w = 0 D[w, {z, 2}]+(Sum[Divide[Subscript[\[Gamma], j],z - Subscript[a, j]], {j, 1, N}])* D[w, z]+Divide[\[CapitalPhi]*(z),Product[z - Subscript[a, j], {j, 1, N}]]*w = 0 Failure Failure Skip Error
31.16#Ex1 ${\displaystyle{\displaystyle x={\sin^{2}}\theta{\cos^{2}}\phi}}$ x = (sin(theta))^(2)* (cos(phi))^(2) x = (Sin[\[Theta]])^(2)* (Cos[\[Phi]])^(2) Failure Failure
Fail
15.36970401+10.48681191*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), x = 1}
16.36970401+10.48681191*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), x = 2}
17.36970401+10.48681191*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), x = 3}
18.74191400+1.298501056*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Error
31.16#Ex2 ${\displaystyle{\displaystyle y={\sin^{2}}\theta{\sin^{2}}\phi}}$ y = (sin(theta))^(2)* (sin(phi))^(2) y = (Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2) Failure Failure
Fail
-17.90774920-11.78531296*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), y = 1}
-16.90774920-11.78531296*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), y = 2}
-15.90774920-11.78531296*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), y = 3}
-21.27995919-0.*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
Error