Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6bm)

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=-23/8

http://functions.wolfram.com/07.23.03.b6bm.01

Input Form

Hypergeometric2F1[-(45/8), -(23/8), 3, z] == (256 2^(1/4) (2 Sqrt[1 - z] (-20730360 + 759249435 z + 37902493597 z^2 + 119772590930 z^3 + 71812030050 z^4 + 4708234895 z^5 - 231307895 z^6 + 9642612 z^7) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (-20730360 + 759249435 z + 37902493597 z^2 + 119772590930 z^3 + 71812030050 z^4 + 4708234895 z^5 - 231307895 z^6 + 9642612 z^7) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 2 (-10365180 + 386102955 z - 20849748754 z^2 - 163421024215 z^3 - 224047443160 z^4 - 61463216915 z^5 - 19514810 z^6 + 803551 z^7) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[1 - z] (-20730360 + 759249435 z + 37902493597 z^2 + 119772590930 z^3 + 71812030050 z^4 + 4708234895 z^5 - 231307895 z^6 + 9642612 z^7) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/(10128586992615 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] 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]