Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8uy)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-11/8

http://functions.wolfram.com/07.23.03.b8uy.01

Input Form

Hypergeometric2F1[-(41/8), -(11/8), 5, z] == (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-16 (29441280 - 479954200 z + 4255453345 z^2 - 34094612310 z^3 - 1426457565675 z^4 - 1651987695618 z^5 - 140223217365 z^6 + 14168490798 z^7 - 1480940769 z^8 + 87191874 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (14720640 - 237561995 z + 2089487510 z^2 - 814738776885 z^3 - 1404167379840 z^4 - 380369923749 z^5 + 9388413342 z^6 - 983833851 z^7 + 58127916 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-14720640 + 229281635 z - 1962176975 z^2 - 1314400655370 z^3 - 1752736200870 z^4 - 184035812961 z^5 + 18575832429 z^6 - 1955211720 z^7 + 116255832 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 8 (29441280 - 479954200 z + 4255453345 z^2 - 34094612310 z^3 - 1426457565675 z^4 - 1651987695618 z^5 - 140223217365 z^6 + 14168490798 z^7 - 1480940769 z^8 + 87191874 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (130748052991681125 Pi 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]