Heun Functions - 31.7 Relations to Other Functions
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 31.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \genhyperF{2}{1}@{\alpha,\beta}{\gamma}{z} = \HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z}}
\genhyperF{2}{1}@{\alpha,\beta}{\gamma}{z} = \HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} | hypergeom([alpha , beta], [gamma], z) = HeunG(1, alpha*beta, alpha, beta, gamma, delta, z)
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Error
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Successful | Missing Macro Error | - | - |
| 31.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z} = \HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z}}
\HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z} = \HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} | HeunG(1, alpha*beta, alpha, beta, gamma, delta, z) = HeunG(0, 0, alpha, beta, gamma, alpha + beta + 1 - gamma, z)
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Error
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Successful | Missing Macro Error | - | - |
| 31.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z} = \HeunHl@{a}{a\alpha\beta}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z}}
\HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z} = \HeunHl@{a}{a\alpha\beta}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} | HeunG(0, 0, alpha, beta, gamma, alpha + beta + 1 - gamma, z) = HeunG(a, a*alpha*beta, alpha, beta, gamma, alpha + beta + 1 - gamma, z)
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Error
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Successful | Missing Macro Error | - | - |
| 31.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}
\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}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} | 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))
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Error
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Failure | Missing Macro Error | Successful [Tested: 90] | - |
| 31.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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)}}
\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)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} | 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))
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Error
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Failure | Missing Macro Error | Successful [Tested: 9] | - |
| 31.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}
\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}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} | 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))
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Error
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Failure | Missing Macro Error | Failed [9 / 9] Result: 0.-.9402251684*I
Test Values: {alpha = 3/2, beta = 3/2, z = 1/2}
Result: 0.-.3436010475*I
Test Values: {alpha = 3/2, beta = 1/2, z = 1/2}
... skip entries to safe data |
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