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

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=-39/8, b>=a

For fixed z and a=-39/8, b=45/8

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

Input Form

Hypergeometric2F1[-(39/8), 45/8, 5, z] == (65536 2^(1/4) (8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-547584 - 2812072 z - 14583702 z^2 - 118121997 z^3 + 2640569701 z^4 - 9138773952 z^5 + 13012699392 z^6 - 8474492928 z^7 + 2095644672 z^8) 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) (-547584 - 2812072 z - 14583702 z^2 - 118121997 z^3 + 2640569701 z^4 - 9138773952 z^5 + 13012699392 z^6 - 8474492928 z^7 + 2095644672 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 4 Sqrt[1 - z] (-547584 - 2812072 z - 14583702 z^2 - 118121997 z^3 + 2640569701 z^4 - 9138773952 z^5 + 13012699392 z^6 - 8474492928 z^7 + 2095644672 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (-2190336 - 10426912 z - 53892105 z^2 - 449339730 z^3 - 13820188613 z^4 + 96186304368 z^5 - 233327533824 z^6 + 269608796160 z^7 - 151658496000 z^8 + 33530314752 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (804989266676595 Pi (1 + Sqrt[1 - z])^(1/4) 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]