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

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=-27/8, b>=a

For fixed z and a=-27/8, b=-15/8

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

Input Form

Hypergeometric2F1[-(27/8), -(15/8), 5, z] == (65536 2^(1/4) (-4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (2889216 - 40780080 z + 311126805 z^2 - 2152860930 z^3 - 26894126410 z^4 - 19829730216 z^5 - 1509502995 z^6 + 30495010 z^7) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (5778432 - 83727072 z + 652246155 z^2 - 4531018635 z^3 + 61014237070 z^4 + 115402803018 z^5 + 27854391927 z^6 + 15247505 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (2889216 - 40780080 z + 311126805 z^2 - 2152860930 z^3 - 26894126410 z^4 - 19829730216 z^5 - 1509502995 z^6 + 30495010 z^7) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 Sqrt[1 - z] (2889216 - 40780080 z + 311126805 z^2 - 2152860930 z^3 - 26894126410 z^4 - 19829730216 z^5 - 1509502995 z^6 + 30495010 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (3710891216267235 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]