View source for 31.8

{{DISPLAYTITLE:Heun Functions - 31.8 Solutions via Quadratures}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta-\alpha = m_{0}+\tfrac{1}{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta - alpha = m[0]+(1)/(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Beta]- \[Alpha] == Subscript[m, 0]+Divide[1,2]</pre></div> || 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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma = -m_{1}+\tfrac{1}{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma = - m[1]+(1)/(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Gamma] == - Subscript[m, 1]+Divide[1,2]</pre></div> || 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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\delta = -m_{2}+\tfrac{1}{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">delta = - m[2]+(1)/(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Delta] == - Subscript[m, 2]+Divide[1,2]</pre></div> || 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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\epsilon = -m_{3}+\tfrac{1}{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">epsilon = - m[3]+(1)/(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Epsilon] == - Subscript[m, 3]+Divide[1,2]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/31.8.E2 31.8.E2] || [[Item:Q9037|<math>w_{+}(\mathbf{m};\lambda;z) = \sqrt{\Psi_{g,N}(\lambda,z)}\*\exp@{+\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\diff{t}}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w_{+}(\mathbf{m};\lambda;z) = \sqrt{\Psi_{g,N}(\lambda,z)}\*\exp@{+\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\diff{t}}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w[+](m ; lambda ; z) = sqrt(Psi[g , N](lambda , z))* exp(+(I*nu(lambda))/(2)*int(((t)^(m[1])*(t - 1)^(m[2])*(t - a)^(m[3]))/(Psi[g , N](lambda , t)*sqrt(t*(t - 1)*(t - a))), t = z[0]..z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[w, +][m ; \[Lambda]; z] == Sqrt[Subscript[\[CapitalPsi], g , N][\[Lambda], z]]* Exp[+Divide[I*\[Nu][\[Lambda]],2]*Integrate[Divide[(t)^(Subscript[m, 1])*(t - 1)^(Subscript[m, 2])*(t - a)^(Subscript[m, 3]),Subscript[\[CapitalPsi], g , N][\[Lambda], t]*Sqrt[t*(t - 1)*(t - a)]], {t, Subscript[z, 0], z}, GenerateConditions->None]]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/31.8.E2 31.8.E2] || [[Item:Q9037|<math>w_{-}(\mathbf{m};\lambda;z) = \sqrt{\Psi_{g,N}(\lambda,z)}\*\exp@{-\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\diff{t}}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w_{-}(\mathbf{m};\lambda;z) = \sqrt{\Psi_{g,N}(\lambda,z)}\*\exp@{-\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\diff{t}}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w[-](m ; lambda ; z) = sqrt(Psi[g , N](lambda , z))* exp(-(I*nu(lambda))/(2)*int(((t)^(m[1])*(t - 1)^(m[2])*(t - a)^(m[3]))/(Psi[g , N](lambda , t)*sqrt(t*(t - 1)*(t - a))), t = z[0]..z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[w, -][m ; \[Lambda]; z] == Sqrt[Subscript[\[CapitalPsi], g , N][\[Lambda], z]]* Exp[-Divide[I*\[Nu][\[Lambda]],2]*Integrate[Divide[(t)^(Subscript[m, 1])*(t - 1)^(Subscript[m, 2])*(t - a)^(Subscript[m, 3]),Subscript[\[CapitalPsi], g , N][\[Lambda], t]*Sqrt[t*(t - 1)*(t - a)]], {t, Subscript[z, 0], z}, GenerateConditions->None]]</syntaxhighlight> || Translation Error || Translation Error || - || -
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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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\Psi_{1,2} = z^{2}+\lambda z+a</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Psi[1 , 2] = (z)^(2)+ lambda*z + a</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[CapitalPsi], 1 , 2] == (z)^(2)+ \[Lambda]*z + a</pre></div> || 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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\nu^{2} = (\lambda+a+1)(\lambda^{2}-4a)</syntaxhighlight> || <math>\mathbf{m} = (1</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(nu)^(2) = (lambda + a + 1)*((lambda)^(2)- 4*a)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Nu]^(2) == (\[Lambda]+ a + 1)*(\[Lambda]^(2)- 4*a)</pre></div> || 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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\Psi_{1,-1} = \left(z^{2}+(\lambda+3a+3)z+a\right)/z^{3}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Psi[1 , - 1] = ((z)^(2)+(lambda + 3*a + 3)*z + a)/(z)^(3)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[CapitalPsi], 1 , - 1] == ((z)^(2)+(\[Lambda]+ 3*a + 3)*z + a)/(z)^(3)</pre></div> || 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>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\nu^{2} = (\lambda+4a+4)\left((\lambda+3a+3)^{2}-4a\right)</syntaxhighlight> || <math>\mathbf{m} = (1</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(nu)^(2) = (lambda + 4*a + 4)*((lambda + 3*a + 3)^(2)- 4*a)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Nu]^(2) == (\[Lambda]+ 4*a + 4)*((\[Lambda]+ 3*a + 3)^(2)- 4*a)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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