Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ba3z)

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=5/8

http://functions.wolfram.com/07.23.03.ba3z.01

Input Form

Hypergeometric2F1[-(39/8), 5/8, 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (303726592 - 3180824192 z + 15364618815 z^2 - 46320946945 z^3 + 106870735790 z^4 + 41859394038 z^5 - 20536556733 z^6 + 7198427115 z^7 - 1545279120 z^8 + 151683840 z^9) EllipticE[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (303726592 - 3180824192 z + 15364618815 z^2 - 46320946945 z^3 + 106870735790 z^4 + 41859394038 z^5 - 20536556733 z^6 + 7198427115 z^7 - 1545279120 z^8 + 151683840 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (303726592 - 3180824192 z + 15364618815 z^2 - 46320946945 z^3 + 106870735790 z^4 + 41859394038 z^5 - 20536556733 z^6 + 7198427115 z^7 - 1545279120 z^8 + 151683840 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (303726592 - 3294721664 z + 16526284047 z^2 - 51773065435 z^3 + 122828060810 z^4 - 419470414902 z^5 + 204042112659 z^6 - 95317807167 z^7 + 31690975140 z^8 - 6471843840 z^9 + 606735360 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (111482385870340485 Pi (1 + Sqrt[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]