Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8qm)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-27/8

http://functions.wolfram.com/07.23.03.b8qm.01

Input Form

Hypergeometric2F1[-(41/8), -(27/8), 5, z] == (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-8 (31173120 - 697823280 z + 9022974345 z^2 - 115183034505 z^3 - 9410297373575 z^4 - 28132625839857 z^5 - 20219465343357 z^6 - 3526214488467 z^7 - 56220074757 z^8 + 901926333 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (7793280 - 173177235 z + 2227721265 z^2 - 2506509478935 z^3 - 9574514145875 z^4 - 9440662068873 z^5 - 2846484900189 z^6 - 214491547149 z^7 + 300642111 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-7793280 + 168793515 z - 2133653940 z^2 - 4102860762495 z^3 - 13786852756570 z^4 - 10766205644787 z^5 - 2030703582864 z^6 - 37379835801 z^7 + 601284222 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 (31173120 - 697823280 z + 9022974345 z^2 - 115183034505 z^3 - 9410297373575 z^4 - 28132625839857 z^5 - 20219465343357 z^6 - 3526214488467 z^7 - 56220074757 z^8 + 901926333 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (406007111921536125 Pi z^4)

Standard Form

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MathML Form

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type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2506509478935 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2227721265 </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'> 173177235 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 7793280 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <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 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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]