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

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=-37/8, b>=a

For fixed z and a=-37/8, b=-23/8

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

Input Form

Hypergeometric2F1[-(37/8), -(23/8), 5, z] == (65536 2^(1/4) (-4 Sqrt[1 - z] (3845632 - 73908240 z + 811300665 z^2 - 8729209090 z^3 - 155952587965 z^4 - 233988153660 z^5 - 69495924945 z^6 - 2038474130 z^7 + 38705205 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 2 Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (3845632 - 73908240 z + 811300665 z^2 - 8729209090 z^3 - 155952587965 z^4 - 233988153660 z^5 - 69495924945 z^6 - 2038474130 z^7 + 38705205 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (7691264 - 152623520 z + 1714197975 z^2 - 18457756730 z^3 + 497870192745 z^4 + 1816129295540 z^5 + 1277495159865 z^6 + 180776193350 z^7 + 12901735 z^8) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 2 Sqrt[1 - z] (3845632 - 73908240 z + 811300665 z^2 - 8729209090 z^3 - 155952587965 z^4 - 233988153660 z^5 - 69495924945 z^6 - 2038474130 z^7 + 38705205 z^8) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (26182397375909775 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] 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]