Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bm3i)

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=-21/8, b>=a

For fixed z and a=-21/8, b=41/8

http://functions.wolfram.com/07.23.03.bm3i.01

Input Form

Hypergeometric2F1[-(21/8), 41/8, 2, z] == (16 2^(1/4) (-((1/Sqrt[1 - z]) (2 (116025 - 20035984 z + 124490432 z^2 - 219660672 z^3 + 116290944 z^4) EllipticE[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])])) - (1/Sqrt[1 - z]) (Sqrt[2 - 2 Sqrt[1 - z]] (116025 - 20035984 z + 124490432 z^2 - 219660672 z^3 + 116290944 z^4) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]) - (-116025 + 6112984 z - 22996688 z^2 + 19381824 z^3) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (1/Sqrt[1 - z]) ((116025 - 20035984 z + 124490432 z^2 - 219660672 z^3 + 116290944 z^4) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])])))/(111035925 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] 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]