Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8un)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-13/8

http://functions.wolfram.com/07.23.03.b8un.01

Input Form

Hypergeometric2F1[-(41/8), -(13/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-1256161280 + 20967098240 z - 178605640125 z^2 + 1115282838450 z^3 - 7205458292175 z^4 - 236026092609324 z^5 - 267279111624843 z^6 - 36435712958190 z^7 + 2151099212535 z^8 - 171605619640 z^9 + 8280126400 z^10) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-1256161280 + 21438158720 z - 186339496365 z^2 + 1180178434875 z^3 - 7606472676825 z^4 - 45954719070849 z^5 + 20618233963641 z^6 + 31424521897953 z^7 + 545634454365 z^8 - 43289535835 z^9 + 2070031600 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-1256161280 + 20967098240 z - 178605640125 z^2 + 1115282838450 z^3 - 7205458292175 z^4 - 236026092609324 z^5 - 267279111624843 z^6 - 36435712958190 z^7 + 2151099212535 z^8 - 171605619640 z^9 + 8280126400 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-1256161280 + 20967098240 z - 178605640125 z^2 + 1115282838450 z^3 - 7205458292175 z^4 - 236026092609324 z^5 - 267279111624843 z^6 - 36435712958190 z^7 + 2151099212535 z^8 - 171605619640 z^9 + 8280126400 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (49145360511815342775 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^5)

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]