Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b78e)

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=-45/8, b>=a

For fixed z and a=-45/8, b=41/8

http://functions.wolfram.com/07.23.03.b78e.01

Input Form

Hypergeometric2F1[-(45/8), 41/8, 5, z] == (65536 2^(1/4) (-2 Sqrt[1 - z] (14283776 + 59366944 z + 270373467 z^2 + 2031950830 z^3 - 141204184717 z^4 + 661577224404 z^5 - 1288384285104 z^6 + 1269539006208 z^7 - 628875582720 z^8 + 124974001152 z^9) EllipticE[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (14283776 + 59366944 z + 270373467 z^2 + 2031950830 z^3 - 141204184717 z^4 + 661577224404 z^5 - 1288384285104 z^6 + 1269539006208 z^7 - 628875582720 z^8 + 124974001152 z^9) EllipticE[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 2 (7141888 + 25219792 z + 115902241 z^2 + 928096715 z^3 - 19976882041 z^4 + 76763270477 z^5 - 132805810272 z^6 + 120030104784 z^7 - 55505852160 z^8 + 10414500096 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[1 - z] (14283776 + 59366944 z + 270373467 z^2 + 2031950830 z^3 - 141204184717 z^4 + 661577224404 z^5 - 1288384285104 z^6 + 1269539006208 z^7 - 628875582720 z^8 + 124974001152 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (3360403781259525 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]