Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ak34)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=5/4, b>=a

For fixed z and a=5/4, b=21/4

http://functions.wolfram.com/07.23.03.ak34.01

Input Form

Hypergeometric2F1[5/4, 21/4, -(3/2), z] == (1/(13260 Pi^(3/2))) (((1/(-1 + z)^8) (4 Sqrt[z] (3315 - 39780 z - 3974094 z^2 - 747516 z^3 + 211827 z^4 - 46200 z^5 + 4928 z^6) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^8) (4 Sqrt[z] (3315 - 39780 z - 3974094 z^2 - 747516 z^3 + 211827 z^4 - 46200 z^5 + 4928 z^6) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7)) ((13260 - 19890 Sqrt[z] - 149175 z + 228735 z^(3/2) + 1481805 z^2 + 6466383 z^(5/2) + 1233771 z^3 + 261261 z^(7/2) - 361053 z^4 - 62601 z^(9/2) + 85008 z^5 + 7392 z^(11/2) - 9856 z^6) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8)) ((-13260 - 19890 Sqrt[z] + 149175 z + 228735 z^(3/2) - 1481805 z^2 + 6466383 z^(5/2) - 1233771 z^3 + 261261 z^(7/2) + 361053 z^4 - 62601 z^(9/2) - 85008 z^5 + 7392 z^(11/2) + 9856 z^6) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 747516 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3974094 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 39780 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3315 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn 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type='integer'> 19890 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 13260 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> 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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]