31.2: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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</syntaxhighlight> || <math>\alpha+\beta+1 = \gamma+\delta+\epsilon</math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
| [https://dlmf.nist.gov/31.2.E1 31.2.E1] || <math qid="Q8976">\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><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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</syntaxhighlight> || <math>\alpha+\beta+1 = \gamma+\delta+\epsilon</math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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| [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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>w(z) = z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">w(z) = (z)^(- gamma/2)*(z - 1)^(- delta/2)*(z - a)^(- epsilon/2)* W(z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">w[z] == (z)^(- \[Gamma]/2)*(z - 1)^(- \[Delta]/2)*(z - a)^(- \[Epsilon]/2)* W[z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/31.2.E2 31.2.E2] || <math qid="Q8977">w(z) = z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>w(z) = z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">w(z) = (z)^(- gamma/2)*(z - 1)^(- delta/2)*(z - a)^(- epsilon/2)* W(z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">w[z] == (z)^(- \[Gamma]/2)*(z - 1)^(- \[Delta]/2)*(z - a)^(- \[Epsilon]/2)* W[z]</pre></div> || 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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z = \sin^{2}@@{\theta}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z = (sin(theta))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == (Sin[\[Theta]])^(2)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2421495608-.799774456e-1*I
| [https://dlmf.nist.gov/31.2.E5 31.2.E5] || <math qid="Q8985">z = \sin^{2}@@{\theta}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z = \sin^{2}@@{\theta}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z = (sin(theta))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == (Sin[\[Theta]])^(2)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2421495608-.799774456e-1*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.123875843+.2860479584*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.123875843+.2860479584*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.24214956105065266, -0.07997744567545023]
Test Values: {theta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.24214956105065266, -0.07997744567545023]
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Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
| [https://dlmf.nist.gov/31.2.E6 31.2.E6] || <math qid="Q8986">\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><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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Latest revision as of 13:11, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
31.2.E1 d 2 w d z 2 + ( γ z + δ z - 1 + ϵ z - a ) ⁢ d w d z + α ⁢ β ⁢ z - q z ⁢ ( z - 1 ) ⁢ ( z - a ) ⁢ w = 0 derivative 𝑤 𝑧 2 𝛾 𝑧 𝛿 𝑧 1 italic-ϵ 𝑧 𝑎 derivative 𝑤 𝑧 𝛼 𝛽 𝑧 𝑞 𝑧 𝑧 1 𝑧 𝑎 𝑤 0 {\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}}
\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		 α + β + 1 = γ + δ + ϵ 𝛼 𝛽 1 𝛾 𝛿 italic-ϵ {\displaystyle{\displaystyle\alpha+\beta+1=\gamma+\delta+\epsilon}} 
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 Manual Skip! Skipped - Because timed out
31.2.E2 w ⁢ ( z ) = z - γ / 2 ⁢ ( z - 1 ) - δ / 2 ⁢ ( z - a ) - ϵ / 2 ⁢ W ⁢ ( z ) 𝑤 𝑧 superscript 𝑧 𝛾 2 superscript 𝑧 1 𝛿 2 superscript 𝑧 𝑎 italic-ϵ 2 𝑊 𝑧 {\displaystyle{\displaystyle w(z)=z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-% \epsilon/2}W(z)}}
w(z) = z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z)		 

w(z) = (z)^(- gamma/2)*(z - 1)^(- delta/2)*(z - a)^(- epsilon/2)* W(z)
 
w[z] == (z)^(- \[Gamma]/2)*(z - 1)^(- \[Delta]/2)*(z - a)^(- \[Epsilon]/2)* W[z]
 Skipped - no semantic math Skipped - no semantic math - -
31.2.E5 z = sin 2 ⁡ θ 𝑧 2 𝜃 {\displaystyle{\displaystyle z={\sin^{2}}\theta}}
z = \sin^{2}@@{\theta}		 

z = (sin(theta))^(2)

z == (Sin[\[Theta]])^(2)
Failure Failure
Failed [70 / 70]
Result: .2421495608-.799774456e-1*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -1.123875843+.2860479584*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[0.24214956105065266, -0.07997744567545023]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1534018595635964, 1.651829143585053]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
31.2.E6 d 2 w d θ 2 + ( ( 2 ⁢ γ - 1 ) ⁢ cot ⁡ θ - ( 2 ⁢ δ - 1 ) ⁢ tan ⁡ θ - ϵ ⁢ sin ⁡ ( 2 ⁢ θ ) a - sin 2 ⁡ θ ) ⁢ d w d θ + 4 ⁢ α ⁢ β ⁢ sin 2 ⁡ θ - q a - sin 2 ⁡ θ ⁢ w = 0 derivative 𝑤 𝜃 2 2 𝛾 1 𝜃 2 𝛿 1 𝜃 italic-ϵ 2 𝜃 𝑎 2 𝜃 derivative 𝑤 𝜃 4 𝛼 𝛽 2 𝜃 𝑞 𝑎 2 𝜃 𝑤 0 {\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}}
\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		 

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 Manual Skip! Skipped - Because timed out
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