Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b5cu)

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=-47/8, b>=a

For fixed z and a=-47/8, b=21/8

http://functions.wolfram.com/07.23.03.b5cu.01

Input Form

Hypergeometric2F1[-(47/8), 21/8, 5, z] == (65536 2^(1/4) (-16 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (12868224 - 54019732 z + 32070027 z^2 + 85151451 z^3 - 590726345 z^4 + 1179934833 z^5 - 1236657114 z^6 + 741176304 z^7 - 241705728 z^8 + 33454080 z^9) EllipticE[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (12868224 - 54019732 z + 32070027 z^2 + 85151451 z^3 - 590726345 z^4 + 1179934833 z^5 - 1236657114 z^6 + 741176304 z^7 - 241705728 z^8 + 33454080 z^9) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 8 Sqrt[1 - z] (12868224 - 54019732 z + 32070027 z^2 + 85151451 z^3 - 590726345 z^4 + 1179934833 z^5 - 1236657114 z^6 + 741176304 z^7 - 241705728 z^8 + 33454080 z^9) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (102945792 - 470762528 z + 408063447 z^2 + 623706732 z^3 + 9223545122 z^4 - 33802495716 z^5 + 54568817919 z^6 - 50171500896 z^7 + 27309959424 z^8 - 8247545856 z^9 + 1070530560 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (465674126442525 Pi (1 + Sqrt[1 - z])^(1/4) z^4)

Standard Form

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MathML Form

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<ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 470762528 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 102945792 </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 /> 4 </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> <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 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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]