Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7ga)

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=-31/8

http://functions.wolfram.com/07.23.03.b7ga.01

Input Form

Hypergeometric2F1[-(43/8), -(31/8), 3, z] == (256 2^(1/4) (-8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (807374106 - 34717086558 z - 2896663718019 z^2 - 13955434379083 z^3 - 15587633935916 z^4 - 4317640630716 z^5 - 171769242827 z^6 + 2174294213 z^7) EllipticE[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (807374106 - 34717086558 z - 2896663718019 z^2 - 13955434379083 z^3 - 15587633935916 z^4 - 4317640630716 z^5 - 171769242827 z^6 + 2174294213 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[1 - z] (807374106 - 34717086558 z - 2896663718019 z^2 - 13955434379083 z^3 - 15587633935916 z^4 - 4317640630716 z^5 - 171769242827 z^6 + 2174294213 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (3229496424 - 140079407391 z + 5859892184478 z^2 + 67945393419107 z^3 + 139706170109308 z^4 + 73870202723415 z^5 + 8440035778846 z^6 + 2174294213 z^7) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(2226534729760341 Pi (1 + Sqrt[1 - z])^(1/4) z^2)

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]