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

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=-29/8, b>=a

For fixed z and a=-29/8, b=7/8

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

Input Form

Hypergeometric2F1[-(29/8), 7/8, 5, z] == (65536 2^(1/4) (-16 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1013376 - 7969780 z + 27735890 z^2 - 57807295 z^3 - 10632225 z^4 + 5472555 z^5 - 1635825 z^6 + 214200 z^7) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 8 Sqrt[1 - z] (1013376 - 7969780 z + 27735890 z^2 - 57807295 z^3 - 10632225 z^4 + 5472555 z^5 - 1635825 z^6 + 214200 z^7) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 8 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1013376 - 7969780 z + 27735890 z^2 - 57807295 z^3 - 10632225 z^4 + 5472555 z^5 - 1635825 z^6 + 214200 z^7) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (8107008 - 66798368 z + 244965175 z^2 - 539569940 z^3 + 621534450 z^4 - 418937460 z^5 + 199475535 z^6 - 55630800 z^7 + 6854400 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (18112129615125 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^4)

Standard Form

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MathML Form

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<cn type='integer'> 539569940 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 244965175 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 66798368 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 8107008 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <apply> <power /> <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> 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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]