Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7di)

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=-43/8, b>=a

For fixed z and a=-43/8, b=-41/8

http://functions.wolfram.com/07.23.03.b7di.01

Input Form

Hypergeometric2F1[-(43/8), -(41/8), 5, z] == (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (128 (-5585184 + 159003207 z - 2702007297 z^2 + 47332165419 z^3 + 5820477941195 z^4 + 28105037221373 z^5 + 38076770914781 z^6 + 17168764282329 z^7 + 2368346378937 z^8 + 65220844920 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (22340736 - 619780887 z + 10359294561 z^2 + 39349283744925 z^3 + 212291030572645 z^4 + 310049438839003 z^5 + 149006451267619 z^6 + 21845759164383 z^7 + 643483834455 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-22340736 + 632347551 z - 10705401432 z^2 + 23906280528900 z^3 + 141016106920280 z^4 + 238527283176602 z^5 + 147065541311768 z^6 + 33583676144964 z^7 + 2436051627240 z^8 + 31323106815 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 64 (-5585184 + 159003207 z - 2702007297 z^2 + 47332165419 z^3 + 5820477941195 z^4 + 28105037221373 z^5 + 38076770914781 z^6 + 17168764282329 z^7 + 2368346378937 z^8 + 65220844920 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (3886068071248988625 Pi 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]