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

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=43/8

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

Input Form

Hypergeometric2F1[-(39/8), 43/8, 1, z] == (2 2^(1/4) (4 Sqrt[1 - z] (156226709 - 2806811792 z + 12450174672 z^2 - 19599849984 z^3 + 10016497920 z^4) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 2 Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (156226709 - 2806811792 z + 12450174672 z^2 - 19599849984 z^3 + 10016497920 z^4) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 2 Sqrt[1 - z] (156226709 - 2806811792 z + 12450174672 z^2 - 19599849984 z^3 + 10016497920 z^4) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (53681351 - 3802130716 z + 31316728976 z^2 - 88012284672 z^3 + 100435695360 z^4 - 40065991680 z^5) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (366134769 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]])

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]