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| |- | | ; Notation : [[31.1|31.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Properties : [[31.2|31.2 Differential Equations]]<br>[[31.3|31.3 Basic Solutions]]<br>[[31.4|31.4 Solutions Analytic at Two Singularities: Heun Functions]]<br>[[31.5|31.5 Solutions Analytic at Three Singularities: Heun Polynomials]]<br>[[31.6|31.6 Path-Multiplicative Solutions]]<br>[[31.7|31.7 Relations to Other Functions]]<br>[[31.8|31.8 Solutions via Quadratures]]<br>[[31.9|31.9 Orthogonality]]<br>[[31.10|31.10 Integral Equations and Representations]]<br>[[31.11|31.11 Expansions in Series of Hypergeometric Functions]]<br>[[31.12|31.12 Confluent Forms of Heun’s Equation]]<br>[[31.13|31.13 Asymptotic Approximations]]<br>[[31.14|31.14 General Fuchsian Equation]]<br>[[31.15|31.15 Stieltjes Polynomials]]<br> |
| |-
| | ; Applications : [[31.16|31.16 Mathematical Applications]]<br>[[31.17|31.17 Physical Applications]]<br> |
| | [https://dlmf.nist.gov/31.2.E1 31.2.E1] || [[Item:Q8976|<math>\deriv[2]{w}{z}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a}\right)\deriv{w}{z}+\frac{\alpha\beta z-q}{z(z-1)(z-a)}w = 0</math>]] || <code>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</code> || <code>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</code> || Failure || Failure || Manual Skip! || Skipped - Because timed out
| | ; Computation : [[31.18|31.18 Methods of Computation]]<br> |
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| | </div> |
| | [https://dlmf.nist.gov/31.2.E2 31.2.E2] || [[Item:Q8977|<math>w(z) = z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z)</math>]] || <code>w*(z) = (z)^(- gamma/ 2)*(z - 1)^(- delta/ 2)*(z - a)^(- epsilon/ 2)* W*(z)</code> || <code>w*(z) == (z)^(- \[Gamma]/ 2)*(z - 1)^(- \[Delta]/ 2)*(z - a)^(- \[Epsilon]/ 2)* W*(z)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.2.E5 31.2.E5] || [[Item:Q8985|<math>z = \sin^{2}@@{\theta}</math>]] || <code>z = (sin(theta))^(2)</code> || <code>z == (Sin[\[Theta]])^(2)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><code>70/70]: [[.2421495608-.799774456e-1*I <- {theta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-1.123875843+.2860479584*I <- {theta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><code>{Complex[0.24214956105065266, -0.07997744567545023] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.1534018595635964, 1.651829143585053] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/31.2.E6 31.2.E6] || [[Item:Q8986|<math>\deriv[2]{w}{\theta}+\left({(2\gamma-1)\cot@@{\theta}-(2\delta-1)\tan@@{\theta}}-\frac{\epsilon\sin@{2\theta}}{a-\sin^{2}@@{\theta}}\right)\deriv{w}{\theta}+4\frac{\alpha\beta\sin^{2}@@{\theta}-q}{a-\sin^{2}@@{\theta}}w = 0</math>]] || <code>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</code> || <code>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</code> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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| | [https://dlmf.nist.gov/31.2.E8 31.2.E8] || [[Item:Q8989|<math>\deriv[2]{w}{\zeta}+\left((2\gamma-1)\frac{\Jacobiellcnk@@{\zeta}{k}\Jacobielldnk@@{\zeta}{k}}{\Jacobiellsnk@@{\zeta}{k}}-(2\delta-1)\frac{\Jacobiellsnk@@{\zeta}{k}\Jacobielldnk@@{\zeta}{k}}{\Jacobiellcnk@@{\zeta}{k}}-(2\epsilon-1)k^{2}\frac{\Jacobiellsnk@@{\zeta}{k}\Jacobiellcnk@@{\zeta}{k}}{\Jacobielldnk@@{\zeta}{k}}\right)\deriv{w}{\zeta}+4k^{2}(\alpha\beta\Jacobiellsnk^{2}@@{\zeta}{k}-q)w = 0</math>]] || <code>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</code> || <code>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</code> || Missing Macro Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/31.3.E1 31.3.E1] || [[Item:Q9003|<math>\HeunHl@{a}{q}{\alpha}{\beta}{\gamma}{\delta}{z} = \sum_{j=0}^{\infty}c_{j}z^{j}</math>]] || <code>HeunG(a, q, alpha, beta, gamma, delta, z) = sum(c[j]*(z)^(j), j = 0..infinity)</code> || <code>Error</code> || Failure || Missing Macro Error || Manual Skip! || -
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| | [https://dlmf.nist.gov/31.3.E2 31.3.E2] || [[Item:Q9004|<math>a\gamma c_{1}-qc_{0} = 0</math>]] || <code>a*gamma*c[1]- q*c[0] = 0</code> || <code>a*\[Gamma]*Subscript[c, 1]- q*Subscript[c, 0] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.3.E3 31.3.E3] || [[Item:Q9005|<math>R_{j}c_{j+1}-(Q_{j}+q)c_{j}+P_{j}c_{j-1} = 0</math>]] || <code>R[j]*c[j + 1]-(Q[j]+ q)* c[j]+ P[j]*c[j - 1] = 0</code> || <code>Subscript[R, j]*Subscript[c, j + 1]-(Subscript[Q, j]+ q)* Subscript[c, j]+ Subscript[P, j]*Subscript[c, j - 1] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.7.E1 31.7.E1] || [[Item:Q9024|<math>\genhyperF{2}{1}@{\alpha,\beta}{\gamma}{z} = \HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z}</math>]] || <code>hypergeom([alpha , beta], [gamma], z) = HeunG(1, alpha*beta, alpha, beta, gamma, delta, z)</code> || <code>Error</code> || Successful || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/31.7.E1 31.7.E1] || [[Item:Q9024|<math>\HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z} = \HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z}</math>]] || <code>HeunG(1, alpha*beta, alpha, beta, gamma, delta, z) = HeunG(0, 0, alpha, beta, gamma, alpha + beta + 1 - gamma, z)</code> || <code>Error</code> || Successful || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/31.7.E1 31.7.E1] || [[Item:Q9024|<math>\HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z} = \HeunHl@{a}{a\alpha\beta}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z}</math>]] || <code>HeunG(0, 0, alpha, beta, gamma, alpha + beta + 1 - gamma, z) = HeunG(a, a*alpha*beta, alpha, beta, gamma, alpha + beta + 1 - gamma, z)</code> || <code>Error</code> || Successful || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/31.7.E2 31.7.E2] || [[Item:Q9025|<math>\HeunHl@{2}{\alpha\beta}{\alpha}{\beta}{\gamma}{\alpha+\beta-2\gamma+1}{z} = \genhyperF{2}{1}@{\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta}{\gamma}{1-(1-z)^{2}}</math>]] || <code>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))</code> || <code>Error</code> || Failure || Missing Macro Error || Successful [Tested: 90] || -
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| | [https://dlmf.nist.gov/31.7.E3 31.7.E3] || [[Item:Q9026|<math>\HeunHl@{4}{\alpha\beta}{\alpha}{\beta}{\tfrac{1}{2}}{\tfrac{2}{3}(\alpha+\beta)}{z} = \genhyperF{2}{1}@{\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta}{\tfrac{1}{2}}{1-(1-z)^{2}(1-\tfrac{1}{4}z)}</math>]] || <code>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))</code> || <code>Error</code> || Failure || Missing Macro Error || Successful [Tested: 9] || -
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| | [https://dlmf.nist.gov/31.7.E4 31.7.E4] || [[Item:Q9027|<math>\HeunHl@{\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} = \genhyperF{2}{1}@{\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}}</math>]] || <code>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))</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><code>9/9]: [[0.-.9402251684*I <- {alpha = 3/2, beta = 3/2, z = 1/2}</code><br><code>0.-.3436010475*I <- {alpha = 3/2, beta = 1/2, z = 1/2}</code><br></div></div> || -
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| | [https://dlmf.nist.gov/31.8#Ex1 31.8#Ex1] || [[Item:Q9033|<math>\beta-\alpha = m_{0}+\tfrac{1}{2}</math>]] || <code>beta - alpha = m[0]+(1)/(2)</code> || <code>\[Beta]- \[Alpha] == Subscript[m, 0]+Divide[1,2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.8#Ex2 31.8#Ex2] || [[Item:Q9034|<math>\gamma = -m_{1}+\tfrac{1}{2}</math>]] || <code>gamma = - m[1]+(1)/(2)</code> || <code>\[Gamma] == - Subscript[m, 1]+Divide[1,2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.8#Ex3 31.8#Ex3] || [[Item:Q9035|<math>\delta = -m_{2}+\tfrac{1}{2}</math>]] || <code>delta = - m[2]+(1)/(2)</code> || <code>\[Delta] == - Subscript[m, 2]+Divide[1,2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.8#Ex4 31.8#Ex4] || [[Item:Q9036|<math>\epsilon = -m_{3}+\tfrac{1}{2}</math>]] || <code>epsilon = - m[3]+(1)/(2)</code> || <code>\[Epsilon] == - Subscript[m, 3]+Divide[1,2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.8#Ex5 31.8#Ex5] || [[Item:Q9039|<math>\Psi_{1,2} = z^{2}+\lambda z+a</math>]] || <code>Psi[1 , 2] = (z)^(2)+ lambda*z + a</code> || <code>Subscript[\[CapitalPsi], 1 , 2] == (z)^(2)+ \[Lambda]*z + a</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.8#Ex6 31.8#Ex6] || [[Item:Q9040|<math>\nu^{2} = (\lambda+a+1)(\lambda^{2}-4a)</math>]] || <code>(nu)^(2) = (lambda + a + 1)*((lambda)^(2)- 4*a)</code> || <code>\[Nu]^(2) == (\[Lambda]+ a + 1)*(\[Lambda]^(2)- 4*a)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.8#Ex7 31.8#Ex7] || [[Item:Q9041|<math>\Psi_{1,-1} = \left(z^{2}+(\lambda+3a+3)z+a\right)/z^{3}</math>]] || <code>Psi[1 , - 1] = ((z)^(2)+(lambda + 3*a + 3)*z + a)/ (z)^(3)</code> || <code>Subscript[\[CapitalPsi], 1 , - 1] == ((z)^(2)+(\[Lambda]+ 3*a + 3)*z + a)/ (z)^(3)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.8#Ex8 31.8#Ex8] || [[Item:Q9042|<math>\nu^{2} = (\lambda+4a+4)\left((\lambda+3a+3)^{2}-4a\right)</math>]] || <code>(nu)^(2) = (lambda + 4*a + 4)*((lambda + 3*a + 3)^(2)- 4*a)</code> || <code>\[Nu]^(2) == (\[Lambda]+ 4*a + 4)*((\[Lambda]+ 3*a + 3)^(2)- 4*a)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.9.E2 31.9.E2] || [[Item:Q9044|<math>\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_{m}(t)w_{k}(t)\diff{t} = \Kroneckerdelta{m}{k}\theta_{m}</math>]] || <code>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]</code> || <code>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 -)}, GenerateConditions->None] == KroneckerDelta[m, k]*Subscript[\[Theta], m]</code> || Translation Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/31.10.E6 31.10.E6] || [[Item:Q9055|<math>p(t) = t^{\gamma}(t-1)^{\delta}(t-a)^{\epsilon}</math>]] || <code>p*(t) = (t)^(gamma)*(t - 1)^(delta)*(t - a)^(epsilon)</code> || <code>p*(t) == (t)^\[Gamma]*(t - 1)^\[Delta]*(t - a)^\[Epsilon]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.10.E8 31.10.E8] || [[Item:Q9059|<math>\sin^{2}@@{\theta}\left(\pderiv[2]{\mathcal{K}}{\theta}+\left((1-2\gamma)\tan@@{\theta}+2(\delta+\epsilon-\tfrac{1}{2})\cot@@{\theta}\right)\pderiv{\mathcal{K}}{\theta}-4\alpha\beta\mathcal{K}\right)+\pderiv[2]{\mathcal{K}}{\phi}+\left((1-2\delta)\cot@@{\phi}-(1-2\epsilon)\tan@@{\phi}\right)\pderiv{\mathcal{K}}{\phi} = 0</math>]] || <code>(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</code> || <code>(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</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-2.252732458-7.327918109*I <- {K = 1/2*3^(1/2)+1/2*I, alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, epsilon = 1}</code><br><code>-2.252732458-7.327918109*I <- {K = 1/2*3^(1/2)+1/2*I, alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, epsilon = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/31.10.E10 31.10.E10] || [[Item:Q9061|<math>\mathcal{K}(z,t) = (zt-a)^{\frac{1}{2}-\delta-\sigma}\*\genhyperF{2}{1}@@{\frac{1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-\sigma+\beta}{\gamma}{\frac{zt}{a}}\*\genhyperF{2}{1}@@{-\frac{1}{2}+\delta+\sigma,-\frac{1}{2}+\epsilon-\sigma}{\delta}{\frac{a(z-1)(t-1)}{(a-1)(zt-a)}}</math>]] || <code>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)))</code> || <code>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)]]</code> || Failure || Failure || Error || Skipped - Because timed out
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| | [https://dlmf.nist.gov/31.10.E18 31.10.E18] || [[Item:Q9071|<math>\pderiv[2]{\mathcal{K}}{u}+\pderiv[2]{\mathcal{K}}{v}+\pderiv[2]{\mathcal{K}}{w}+\frac{2\gamma-1}{u}\pderiv{\mathcal{K}}{u}+\frac{2\delta-1}{v}\pderiv{\mathcal{K}}{v}+\frac{2\epsilon-1}{w}\pderiv{\mathcal{K}}{w} = 0</math>]] || <code>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</code> || <code>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</code> || Successful || Successful || - || -
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| | [https://dlmf.nist.gov/31.10#Ex7 31.10#Ex7] || [[Item:Q9073|<math>u = r\cos@@{\theta}</math>]] || <code>u = r*cos(theta)</code> || <code>u == r*Cos[\[Theta]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[1.961839932-.954243254e-1*I <- {r = -3/2, theta = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I}</code><br><code>.5958145280+.2706010786*I <- {r = -3/2, theta = 1/2*3^(1/2)+1/2*I, u = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[1.9618399323702764, -0.09542432534354878] <- {Rule[r, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[2.7076736790806044, 1.2036130644027554] <- {Rule[r, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/31.10#Ex8 31.10#Ex8] || [[Item:Q9074|<math>v = r\sin@@{\theta}\sin@@{\phi}</math>]] || <code>v = r*sin(theta)*sin(phi)</code> || <code>v == r*Sin[\[Theta]]*Sin[\[Phi]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[1.801839169+1.369966168*I <- {phi = 1/2*3^(1/2)+1/2*I, r = -3/2, theta = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I}</code><br><code>.4358137648+1.735991572*I <- {phi = 1/2*3^(1/2)+1/2*I, r = -3/2, theta = 1/2*3^(1/2)+1/2*I, v = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[1.801839167885118, 1.3699661685131752] <- {Rule[r, -1.5], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.4330012446224153, 1.2666732793219693] <- {Rule[r, -1.5], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/31.10#Ex9 31.10#Ex9] || [[Item:Q9075|<math>w = r\sin@@{\theta}\cos@@{\phi}</math>]] || <code>w = r*sin(theta)*cos(phi)</code> || <code>w == r*Sin[\[Theta]]*Cos[\[Phi]]</code> || Failure || Failure || Manual Skip! || Skipped - Because timed out
| |
| |-
| |
| | [https://dlmf.nist.gov/31.10.E21 31.10.E21] || [[Item:Q9076|<math>\pderiv[2]{\mathcal{K}}{r}+\frac{2(\gamma+\delta+\epsilon)-1}{r}\pderiv{\mathcal{K}}{r}+\frac{1}{r^{2}}\pderiv[2]{\mathcal{K}}{\theta}+\frac{(2(\delta+\epsilon)-1)\cot@@{\theta}-(2\gamma-1)\tan@@{\theta}}{r^{2}}\pderiv{\mathcal{K}}{\theta}+\frac{1}{r^{2}\sin^{2}@@{\theta}}\pderiv[2]{\mathcal{K}}{\phi}+\frac{(2\delta-1)\cot@@{\phi}-(2\epsilon-1)\tan@@{\phi}}{r^{2}\sin^{2}@@{\theta}}\pderiv{\mathcal{K}}{\phi} = 0</math>]] || <code>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</code> || <code>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</code> || Successful || Successful || - || Successful [Tested: 300]
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E1 31.11.E1] || [[Item:Q9085|<math>w(z) = \sum_{j=0}^{\infty}c_{j}P_{j}</math>]] || <code>w*(z) = sum(c[j]*P[j], j = 0..infinity)</code> || <code>w*(z) == Sum[Subscript[c, j]*Subscript[P, j], {j, 0, Infinity}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E4 31.11.E4] || [[Item:Q9088|<math>L_{0}c_{0}+M_{0}c_{1} = 0</math>]] || <code>L[0]*c[0]+ M[0]*c[1] = 0</code> || <code>Subscript[L, 0]*Subscript[c, 0]+ Subscript[M, 0]*Subscript[c, 1] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E5 31.11.E5] || [[Item:Q9089|<math>K_{j}c_{j-1}+L_{j}c_{j}+M_{j}c_{j+1} = 0</math>]] || <code>K[j]*c[j - 1]+ L[j]*c[j]+ M[j]*c[j + 1] = 0</code> || <code>Subscript[K, j]*Subscript[c, j - 1]+ Subscript[L, j]*Subscript[c, j]+ Subscript[M, j]*Subscript[c, j + 1] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E6 31.11.E6] || [[Item:Q9090|<math>K_{j} = -\frac{(j+\alpha-\mu-1)(j+\beta-\mu-1)(j+\gamma-\mu-1)(j+\lambda-1)}{(2j+\lambda-\mu-1)(2j+\lambda-\mu-2)}</math>]] || <code>K[j] = -((j + alpha - mu - 1)*(j + beta - mu - 1)*(j + gamma - mu - 1)*(j + lambda - 1))/((2*j + lambda - mu - 1)*(2*j + lambda - mu - 2))</code> || <code>Subscript[K, j] == -Divide[(j + \[Alpha]- \[Mu]- 1)*(j + \[Beta]- \[Mu]- 1)*(j + \[Gamma]- \[Mu]- 1)*(j + \[Lambda]- 1),(2*j + \[Lambda]- \[Mu]- 1)*(2*j + \[Lambda]- \[Mu]- 2)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E7 31.11.E7] || [[Item:Q9091|<math>L_{j} = a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma-\mu)(j+\lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)(j-\beta+\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)}</math>]] || <code>L[j] = a*(lambda + j)*(mu - j)- q +((j + alpha - mu)*(j + beta - mu)*(j + gamma - mu)*(j + lambda))/((2*j + lambda - mu)*(2*j + lambda - mu + 1))+((j - alpha + lambda)*(j - beta + lambda)*(j - gamma + lambda)*(j - mu))/((2*j + lambda - mu)*(2*j + lambda - mu - 1))</code> || <code>Subscript[L, j] == a*(\[Lambda]+ j)*(\[Mu]- j)- q +Divide[(j + \[Alpha]- \[Mu])*(j + \[Beta]- \[Mu])*(j + \[Gamma]- \[Mu])*(j + \[Lambda]),(2*j + \[Lambda]- \[Mu])*(2*j + \[Lambda]- \[Mu]+ 1)]+Divide[(j - \[Alpha]+ \[Lambda])*(j - \[Beta]+ \[Lambda])*(j - \[Gamma]+ \[Lambda])*(j - \[Mu]),(2*j + \[Lambda]- \[Mu])*(2*j + \[Lambda]- \[Mu]- 1)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E8 31.11.E8] || [[Item:Q9092|<math>M_{j} = -\frac{(j-\alpha+\lambda+1)(j-\beta+\lambda+1)(j-\gamma+\lambda+1)(j-\mu+1)}{(2j+\lambda-\mu+1)(2j+\lambda-\mu+2)}</math>]] || <code>M[j] = -((j - alpha + lambda + 1)*(j - beta + lambda + 1)*(j - gamma + lambda + 1)*(j - mu + 1))/((2*j + lambda - mu + 1)*(2*j + lambda - mu + 2))</code> || <code>Subscript[M, j] == -Divide[(j - \[Alpha]+ \[Lambda]+ 1)*(j - \[Beta]+ \[Lambda]+ 1)*(j - \[Gamma]+ \[Lambda]+ 1)*(j - \[Mu]+ 1),(2*j + \[Lambda]- \[Mu]+ 1)*(2*j + \[Lambda]- \[Mu]+ 2)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E9 31.11.E9] || [[Item:Q9093|<math>M_{-1}P_{-1} = 0</math>]] || <code>M[- 1]*P[- 1] = 0</code> || <code>Subscript[M, - 1]*Subscript[P, - 1] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex1 31.11#Ex1] || [[Item:Q9094|<math>\lambda = \alpha</math>]] || <code>lambda = alpha</code> || <code>\[Lambda] == \[Alpha]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex2 31.11#Ex2] || [[Item:Q9095|<math>\mu = \beta-\epsilon</math>]] || <code>mu = beta - epsilon</code> || <code>\[Mu] == \[Beta]- \[Epsilon]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex3 31.11#Ex3] || [[Item:Q9096|<math>\lambda = \beta</math>]] || <code>lambda = beta</code> || <code>\[Lambda] == \[Beta]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex4 31.11#Ex4] || [[Item:Q9097|<math>\mu = \alpha-\epsilon</math>]] || <code>mu = alpha - epsilon</code> || <code>\[Mu] == \[Alpha]- \[Epsilon]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11.E12 31.11.E12] || [[Item:Q9098|<math>P_{j} = \frac{\EulerGamma@{\alpha+j}\EulerGamma@{1-\gamma+\alpha+j}}{\EulerGamma@{1+\alpha-\beta+\epsilon+2j}}z^{-\alpha-j}\*\genhyperF{2}{1}@@{\alpha+j,1-\gamma+\alpha+j}{1+\alpha-\beta+\epsilon+2j}{\frac{1}{z}}</math>]] || <code>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))</code> || <code>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]]</code> || Missing Macro Error || Missing Macro Error || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex5 31.11#Ex5] || [[Item:Q9100|<math>\lambda = \gamma+\delta-1</math>]] || <code>lambda = gamma + delta - 1</code> || <code>\[Lambda] == \[Gamma]+ \[Delta]- 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex6 31.11#Ex6] || [[Item:Q9101|<math>\mu = 0</math>]] || <code>mu = 0</code> || <code>\[Mu] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex7 31.11#Ex7] || [[Item:Q9102|<math>\lambda = \gamma</math>]] || <code>lambda = gamma</code> || <code>\[Lambda] == \[Gamma]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex8 31.11#Ex8] || [[Item:Q9103|<math>\mu = \delta-1</math>]] || <code>mu = delta - 1</code> || <code>\[Mu] == \[Delta]- 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex9 31.11#Ex9] || [[Item:Q9104|<math>\lambda = \delta</math>]] || <code>lambda = delta</code> || <code>\[Lambda] == \[Delta]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex10 31.11#Ex10] || [[Item:Q9105|<math>\mu = \gamma-1</math>]] || <code>mu = gamma - 1</code> || <code>\[Mu] == \[Gamma]- 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex11 31.11#Ex11] || [[Item:Q9106|<math>\lambda = 1</math>]] || <code>lambda = 1</code> || <code>\[Lambda] == 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.11#Ex12 31.11#Ex12] || [[Item:Q9107|<math>\mu = \gamma+\delta-2</math>]] || <code>mu = gamma + delta - 2</code> || <code>\[Mu] == \[Gamma]+ \[Delta]- 2</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/31.12.E1 31.12.E1] || [[Item:Q9108|<math>\deriv[2]{w}{z}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon\right)\deriv{w}{z}+\frac{\alpha z-q}{z(z-1)}w = 0</math>]] || <code>diff(w, [z$(2)])+((gamma)/(z)+(delta)/(z - 1)+ epsilon)* diff(w, z)+(alpha*z - q)/(z*(z - 1))*w = 0</code> || <code>D[w, {z, 2}]+(Divide[\[Gamma],z]+Divide[\[Delta],z - 1]+ \[Epsilon])* D[w, z]+Divide[\[Alpha]*z - q,z*(z - 1)]*w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.2500000003-.9330127021*I <- {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, epsilon = 1}</code><br><code>.2500000003-.9330127021*I <- {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, epsilon = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.25, -0.9330127018922194] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϵ, 1]}</code><br><code>Complex[0.25, -0.9330127018922194] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϵ, 2]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/31.12.E2 31.12.E2] || [[Item:Q9109|<math>\deriv[2]{w}{z}+\left(\frac{\delta}{z^{2}}+\frac{\gamma}{z}+1\right)\deriv{w}{z}+\frac{\alpha z-q}{z^{2}}w = 0</math>]] || <code>diff(w, [z$(2)])+((delta)/((z)^(2))+(gamma)/(z)+ 1)* diff(w, z)+(alpha*z - q)/((z)^(2))*w = 0</code> || <code>D[w, {z, 2}]+(Divide[\[Delta],(z)^(2)]+Divide[\[Gamma],z]+ 1)* D[w, z]+Divide[\[Alpha]*z - q,(z)^(2)]*w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.5000000000+0.*I <- {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>.9999999998-1.500000000*I <- {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.49999999999999994, -2.7755575615628914*^-17] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.49999999999999994, -2.7755575615628914*^-17] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/31.12.E3 31.12.E3] || [[Item:Q9110|<math>\deriv[2]{w}{z}-\left(\frac{\gamma}{z}+\delta+z\right)\deriv{w}{z}+\frac{\alpha z-q}{z}w = 0</math>]] || <code>diff(w, [z$(2)])-((gamma)/(z)+ delta + z)* diff(w, z)+(alpha*z - q)/(z)*w = 0</code> || <code>D[w, {z, 2}]-(Divide[\[Gamma],z]+ \[Delta]+ z)* D[w, z]+Divide[\[Alpha]*z - q,z]*w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.4330127020+.2500000000*I <- {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>.7990381058+1.616025404*I <- {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.4330127018922193, 0.24999999999999994] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.4330127018922193, 0.24999999999999994] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/31.12.E4 31.12.E4] || [[Item:Q9111|<math>\deriv[2]{w}{z}+\left(\gamma+z\right)z\deriv{w}{z}+\left(\alpha z-q\right)w = 0</math>]] || <code>diff(w, [z$(2)])+(gamma + z)* z*diff(w, z)+(alpha*z - q)* w = 0</code> || <code>D[w, {z, 2}]+(\[Gamma]+ z)* z*D[w, z]+(\[Alpha]*z - q)* w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.2500000002+.4330127020*I <- {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-1.799038106-.1160254034*I <- {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.25000000000000006, 0.43301270189221924] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.25000000000000006, 0.43301270189221924] <- {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/31.14.E1 31.14.E1] || [[Item:Q9112|<math>\deriv[2]{w}{z}+\left(\sum_{j=1}^{N}\frac{\gamma_{j}}{z-a_{j}}\right)\deriv{w}{z}+\left(\sum_{j=1}^{N}\frac{q_{j}}{z-a_{j}}\right)w = 0</math>]] || <code>diff(w, [z$(2)])+(sum((gamma[j])/(z - a[j]), j = 1..N))* diff(w, z)+(sum((q[j])/(z - a[j]), j = 1..N))* w = 0</code> || <code>D[w, {z, 2}]+(Sum[Divide[Subscript[\[Gamma], j],z - Subscript[a, j]], {j, 1, N}, GenerateConditions->None])* D[w, z]+(Sum[Divide[Subscript[q, j],z - Subscript[a, j]], {j, 1, N}, GenerateConditions->None])* w == 0</code> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
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| | [https://dlmf.nist.gov/31.14#Ex1 31.14#Ex1] || [[Item:Q9113|<math>\alpha+\beta+1 = \sum_{j=1}^{N}\gamma_{j}</math>]] || <code>alpha + beta + 1 = sum(gamma[j], j = 1..N)</code> || <code>\[Alpha]+ \[Beta]+ 1 == Sum[Subscript[\[Gamma], j], {j, 1, N}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.14#Ex2 31.14#Ex2] || [[Item:Q9114|<math>\alpha\beta = \sum_{j=1}^{N}a_{j}q_{j}</math>]] || <code>alpha*beta = sum(a[j]*q[j], j = 1..N)</code> || <code>\[Alpha]*\[Beta] == Sum[Subscript[a, j]*Subscript[q, j], {j, 1, N}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.14.E3 31.14.E3] || [[Item:Q9115|<math>w(z) = \left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z)</math>]] || <code>w*(z) = (product((z - a[j])^(- gamma[j]/ 2), j = 1..N))* W*(z)</code> || <code>w*(z) == (Product[(z - Subscript[a, j])^(- Subscript[\[Gamma], j]/ 2), {j, 1, N}, GenerateConditions->None])* W*(z)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.15.E1 31.15.E1] || [[Item:Q9119|<math>\deriv[2]{w}{z}+\left(\sum_{j=1}^{N}\frac{\gamma_{j}}{z-a_{j}}\right)\deriv{w}{z}+\frac{\Phi(z)}{\prod_{j=1}^{N}(z-a_{j})}w = 0</math>]] || <code>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</code> || <code>D[w, {z, 2}]+(Sum[Divide[Subscript[\[Gamma], j],z - Subscript[a, j]], {j, 1, N}, GenerateConditions->None])* D[w, z]+Divide[\[CapitalPhi]*(z),Product[z - Subscript[a, j], {j, 1, N}, GenerateConditions->None]]*w == 0</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Times[Complex[0.0, 1.0], Power[NProduct[0 <- {j, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]}, Rule[GenerateConditions, None]], -1]], {Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Φ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Times[Complex[0.0, 1.0], Power[NProduct[0 <- {j, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]}, Rule[GenerateConditions, None]], -1]], {Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Φ, Power[E, Times[Comp</div></div>
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| | [https://dlmf.nist.gov/31.15#Ex1 31.15#Ex1] || [[Item:Q9123|<math>\gamma_{j} > 0</math>]] || <code>gamma[j] > 0</code> || <code>Subscript[\[Gamma], j] > 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.15.E6 31.15.E6] || [[Item:Q9125|<math>a_{j} < a_{j+1}</math>]] || <code>a[j] < a[j + 1]</code> || <code>Subscript[a, j] < Subscript[a, j + 1]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.15.E7 31.15.E7] || [[Item:Q9126|<math>q_{j} = \gamma_{j}\sum_{k=1}^{n}\frac{1}{z_{k}-a_{j}}</math>]] || <code>q[j] = gamma[j]*sum((1)/(z[k]- a[j]), k = 1..n)</code> || <code>Subscript[q, j] == Subscript[\[Gamma], j]*Sum[Divide[1,Subscript[z, k]- Subscript[a, j]], {k, 1, n}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.16.E4 31.16.E4] || [[Item:Q9137|<math>P_{j}A_{j-1}+Q_{j}A_{j}+R_{j}A_{j+1} = 0</math>]] || <code>P[j]*A[j - 1]+ Q[j]*A[j]+ R[j]*A[j + 1] = 0</code> || <code>Subscript[P, j]*Subscript[A, j - 1]+ Subscript[Q, j]*Subscript[A, j]+ Subscript[R, j]*Subscript[A, j + 1] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17.E2 31.17.E2] || [[Item:Q9143|<math>\frac{x_{s}^{2}}{z_{k}}+\frac{x_{t}^{2}}{z_{k}-1}+\frac{x_{u}^{2}}{z_{k}-a} = 0</math>]] || <code>(x(x[s])^(2))/(x + y*I[k])(x(x[t])^(2))/(x + y*I[k]- 1)(x(x[u])^(2))/(x + y*I[k]- a) = 0</code> || <code>Divide[x(Subscript[x, s])^(2),Subscript[x + y*I, k]]Divide[x(Subscript[x, t])^(2),Subscript[x + y*I, k]- 1]Divide[x(Subscript[x, u])^(2),Subscript[x + y*I, k]- a] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17.E4 31.17.E4] || [[Item:Q9147|<math>\Psi(\mathbf{x}) = (z_{1}z_{2})^{-s-\frac{1}{4}}((z_{1}-1)(z_{2}-1))^{-t-\frac{1}{4}}\*((z_{1}-a)(z_{2}-a))^{-u-\frac{1}{4}}w(z_{1})w(z_{2})</math>]] || <code>Psi*(x) = (x + y*I[1]*x + y*I[2])^(- s -(1)/(4))*((x + y*I[1]- 1)*(x + y*I[2]- 1))^(- t -(1)/(4))*((x + y*I[1]- a)*(x + y*I[2]- a))^(- u -(1)/(4))* w*(x + y*I[1])* w*(x + y*I[2])</code> || <code>\[CapitalPsi]*(x) == (Subscript[x + y*I, 1]*Subscript[x + y*I, 2])^(- s -Divide[1,4])*((Subscript[x + y*I, 1]- 1)*(Subscript[x + y*I, 2]- 1))^(- t -Divide[1,4])*((Subscript[x + y*I, 1]- a)*(Subscript[x + y*I, 2]- a))^(- u -Divide[1,4])* w*(Subscript[x + y*I, 1])* w*(Subscript[x + y*I, 2])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17#Ex6 31.17#Ex6] || [[Item:Q9148|<math>\alpha = -s-t-u-j-1</math>]] || <code>alpha = - s - t - u - j - 1</code> || <code>\[Alpha] == - s - t - u - j - 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17#Ex7 31.17#Ex7] || [[Item:Q9149|<math>\beta = j-s-t-u</math>]] || <code>beta = j - s - t - u</code> || <code>\[Beta] == j - s - t - u</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17#Ex8 31.17#Ex8] || [[Item:Q9150|<math>\gamma = -2s</math>]] || <code>gamma = - 2*s</code> || <code>\[Gamma] == - 2*s</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17#Ex9 31.17#Ex9] || [[Item:Q9151|<math>\delta = -2t</math>]] || <code>delta = - 2*t</code> || <code>\[Delta] == - 2*t</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17#Ex10 31.17#Ex10] || [[Item:Q9152|<math>\epsilon = -2u</math>]] || <code>epsilon = - 2*u</code> || <code>\[Epsilon] == - 2*u</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/31.17#Ex11 31.17#Ex11] || [[Item:Q9153|<math>q = ah_{s}+2s(at+u)</math>]] || <code>q = a*h[s]+ 2*s*(a*t + u)</code> || <code>q == a*Subscript[h, s]+ 2*s*(a*t + u)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |}
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