Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9746)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-23/4, b>=a

For fixed z and a=-23/4, b=5/2

http://functions.wolfram.com/07.23.03.9746.01

Input Form

Hypergeometric2F1[-(23/4), 5/2, 1, z] == (2 (1 - z)^(1/4) (2819536 - 26855129 z + 80827588 z^2 - 109109910 z^3 + 69150900 z^4 - 16823625 z^5) EllipticE[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + 2 (1 - z)^(3/4) (2819536 - 26855129 z + 80827588 z^2 - 109109910 z^3 + 69150900 z^4 - 16823625 z^5) EllipticE[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (1 - z)^(1/4) (-2819536 + 26855129 z - 80827588 z^2 + 109109910 z^3 - 69150900 z^4 + 16823625 z^5) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + Sqrt[1 - z] (-2819536 + 26855129 z - 80827588 z^2 + 109109910 z^3 - 69150900 z^4 + 16823625 z^5) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (1 - z)^(3/4) (-2819536 + 26855129 z - 80827588 z^2 + 109109910 z^3 - 69150900 z^4 + 16823625 z^5) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (-800596 - 1009733 z + 30173989 z^2 - 92910610 z^3 + 120248310 z^4 - 72515625 z^5 + 16823625 z^6) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)])/ (504735 Sqrt[2] Pi 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]