Heun Functions - 31.7 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
31.7.E1 F 1 2 ⁡ ( α , β ; γ ; z ) = H ⁢ ℓ ⁡ ( 1 , α ⁢ β ; α , β , γ , δ ; z ) Gauss-hypergeometric-F-as-2F1 𝛼 𝛽 𝛾 𝑧 Heun-local 1 𝛼 𝛽 𝛼 𝛽 𝛾 𝛿 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=% \mathit{H\!\ell}\left(1,\alpha\beta;\alpha,\beta,\gamma,\delta;z\right)}}
\genhyperF{2}{1}@{\alpha,\beta}{\gamma}{z} = \HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
hypergeom([alpha , beta], [gamma], z) = HeunG(1, alpha*beta, alpha, beta, gamma, delta, z)

Error
Successful Missing Macro Error - -
31.7.E1 H ⁢ ℓ ⁡ ( 1 , α ⁢ β ; α , β , γ , δ ; z ) = H ⁢ ℓ ⁡ ( 0 , 0 ; α , β , γ , α + β + 1 - γ ; z ) Heun-local 1 𝛼 𝛽 𝛼 𝛽 𝛾 𝛿 𝑧 Heun-local 0 0 𝛼 𝛽 𝛾 𝛼 𝛽 1 𝛾 𝑧 {\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)}}
\HeunHl@{1}{\alpha\beta}{\alpha}{\beta}{\gamma}{\delta}{z} = \HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
HeunG(1, alpha*beta, alpha, beta, gamma, delta, z) = HeunG(0, 0, alpha, beta, gamma, alpha + beta + 1 - gamma, z)

Error
Successful Missing Macro Error - -
31.7.E1 H ⁢ ℓ ⁡ ( 0 , 0 ; α , β , γ , α + β + 1 - γ ; z ) = H ⁢ ℓ ⁡ ( a , a ⁢ α ⁢ β ; α , β , γ , α + β + 1 - γ ; z ) Heun-local 0 0 𝛼 𝛽 𝛾 𝛼 𝛽 1 𝛾 𝑧 Heun-local 𝑎 𝑎 𝛼 𝛽 𝛼 𝛽 𝛾 𝛼 𝛽 1 𝛾 𝑧 {\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)}}
\HeunHl@{0}{0}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z} = \HeunHl@{a}{a\alpha\beta}{\alpha}{\beta}{\gamma}{\alpha+\beta+1-\gamma}{z}		 | z | < 1 𝑧 1 {\displaystyle{\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)

Error
Successful Missing Macro Error - -
31.7.E2 H ⁢ ℓ ⁡ ( 2 , α ⁢ β ; α , β , γ , α + β - 2 ⁢ γ + 1 ; z ) = F 1 2 ⁡ ( 1 2 ⁢ α , 1 2 ⁢ β ; γ ; 1 - ( 1 - z ) 2 ) Heun-local 2 𝛼 𝛽 𝛼 𝛽 𝛾 𝛼 𝛽 2 𝛾 1 𝑧 Gauss-hypergeometric-F-as-2F1 1 2 𝛼 1 2 𝛽 𝛾 1 superscript 1 𝑧 2 {\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)}}
\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}}		 | z | < 1 𝑧 1 {\displaystyle{\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))

Error
Failure Missing Macro Error Successful [Tested: 90] -
31.7.E3 H ⁢ ℓ ⁡ ( 4 , α ⁢ β ; α , β , 1 2 , 2 3 ⁢ ( α + β ) ; z ) = F 1 2 ⁡ ( 1 3 ⁢ α , 1 3 ⁢ β ; 1 2 ; 1 - ( 1 - z ) 2 ⁢ ( 1 - 1 4 ⁢ z ) ) Heun-local 4 𝛼 𝛽 𝛼 𝛽 1 2 2 3 𝛼 𝛽 𝑧 Gauss-hypergeometric-F-as-2F1 1 3 𝛼 1 3 𝛽 1 2 1 superscript 1 𝑧 2 1 1 4 𝑧 {\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)}}
\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)}		 | z | < 1 𝑧 1 {\displaystyle{\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))

Error
Failure Missing Macro Error Successful [Tested: 9] -
31.7.E4 H ⁢ ℓ ⁡ ( 1 2 + i ⁢ 3 2 , α ⁢ β ⁢ ( 1 2 + i ⁢ 3 6 ) ; α , β , 1 3 ⁢ ( α + β + 1 ) , 1 3 ⁢ ( α + β + 1 ) ; z ) = F 1 2 ⁡ ( 1 3 ⁢ α , 1 3 ⁢ β ; 1 3 ⁢ ( α + β + 1 ) ; 1 - ( 1 - ( 3 2 - i ⁢ 3 2 ) ⁢ z ) 3 ) Heun-local 1 2 𝑖 3 2 𝛼 𝛽 1 2 𝑖 3 6 𝛼 𝛽 1 3 𝛼 𝛽 1 1 3 𝛼 𝛽 1 𝑧 Gauss-hypergeometric-F-as-2F1 1 3 𝛼 1 3 𝛽 1 3 𝛼 𝛽 1 1 superscript 1 3 2 𝑖 3 2 𝑧 3 {\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)}}
\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}}		 | z | < 1 𝑧 1 {\displaystyle{\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))

Error
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}

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