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

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-37/8, b>=a

For fixed z and a=-37/8, b=-25/8

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

Input Form

Hypergeometric2F1[-(37/8), -(25/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (151765120 - 3532096035 z + 59892681810 z^2 + 5092777121131 z^3 + 15505094653020 z^4 + 10227060299715 z^5 + 1361642588530 z^6 + 205283925 z^7) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (151765120 - 3532096035 z + 59892681810 z^2 + 5092777121131 z^3 + 15505094653020 z^4 + 10227060299715 z^5 + 1361642588530 z^6 + 205283925 z^7) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (151765120 - 3532096035 z + 59892681810 z^2 + 5092777121131 z^3 + 15505094653020 z^4 + 10227060299715 z^5 + 1361642588530 z^6 + 205283925 z^7) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (151765120 - 3589007955 z + 61201655970 z^2 + 1348015469071 z^3 + 1359842898624 z^4 - 1674075409125 z^5 - 1034273156630 z^6 - 58095350775 z^7 + 821135700 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (3811999546329975 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^3)

Standard Form

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MathML Form

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type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn 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</semantics> </math>

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]