Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.be02)

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=-33/8, b>=a

For fixed z and a=-33/8, b=3/8

http://functions.wolfram.com/07.23.03.be02.01

Input Form

Hypergeometric2F1[-(33/8), 3/8, 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (275742720 - 2887758720 z + 14063891325 z^2 - 43499645475 z^3 + 107744229450 z^4 + 549397908514 z^5 - 159381985399 z^6 + 47496972705 z^7 - 9381539440 z^8 + 874881280 z^9) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (275742720 - 2991162240 z + 15118526445 z^2 - 48492517425 z^3 + 122761975050 z^4 - 54924138386 z^5 - 41875274623 z^6 + 12294523059 z^7 - 2386394920 z^8 + 218720320 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (275742720 - 2887758720 z + 14063891325 z^2 - 43499645475 z^3 + 107744229450 z^4 + 549397908514 z^5 - 159381985399 z^6 + 47496972705 z^7 - 9381539440 z^8 + 874881280 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (275742720 - 2887758720 z + 14063891325 z^2 - 43499645475 z^3 + 107744229450 z^4 + 549397908514 z^5 - 159381985399 z^6 + 47496972705 z^7 - 9381539440 z^8 + 874881280 z^9) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (148821264070253775 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^5)

Standard Form

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MathML Form

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type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 14063891325 </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'> 2887758720 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 275742720 </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> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> 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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]