Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bbk7)

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=-1/8

http://functions.wolfram.com/07.23.03.bbk7.01

Input Form

Hypergeometric2F1[-(37/8), -(1/8), 4, z] == (2048 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-12498304 + 147929145 z - 1036968660 z^2 - 20539401170 z^3 - 252740700 z^4 + 78670305 z^5 - 16819800 z^6 + 1713600 z^7) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-12498304 + 147929145 z - 1036968660 z^2 - 20539401170 z^3 - 252740700 z^4 + 78670305 z^5 - 16819800 z^6 + 1713600 z^7) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-12498304 + 147929145 z - 1036968660 z^2 - 20539401170 z^3 - 252740700 z^4 + 78670305 z^5 - 16819800 z^6 + 1713600 z^7) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-12498304 + 152616009 z - 1091160525 z^2 - 2477399030 z^3 + 4300914150 z^4 - 1154953395 z^5 + 346190295 z^6 - 70563600 z^7 + 6854400 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (18112129615125 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^3)

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]