Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bb8l)

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=-31/8

http://functions.wolfram.com/07.23.03.bb8l.01

Input Form

Hypergeometric2F1[-(37/8), -(31/8), 5, z] == (65536 2^(1/4) (2 Sqrt[1 - z] (-34061312 + 752542112 z - 9698991855 z^2 + 125904113930 z^3 + 3022591168495 z^4 + 6498769864620 z^5 + 3375467789535 z^6 + 394773891210 z^7 + 3599584065 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (-34061312 + 752542112 z - 9698991855 z^2 + 125904113930 z^3 + 3022591168495 z^4 + 6498769864620 z^5 + 3375467789535 z^6 + 394773891210 z^7 + 3599584065 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[1 - z] (-34061312 + 752542112 z - 9698991855 z^2 + 125904113930 z^3 + 3022591168495 z^4 + 6498769864620 z^5 + 3375467789535 z^6 + 394773891210 z^7 + 3599584065 z^8) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 4 (8515328 - 193457608 z + 2541459515 z^2 - 32972614010 z^3 + 1061587442245 z^4 + 5210430245740 z^5 + 5516338778325 z^6 + 1563805751670 z^7 + 90579779595 z^8) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (235641576383187975 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^4)

Standard Form

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MathML Form

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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]