Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9nn)

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=-19/4, b>=a

For fixed z and a=-19/4, b=13/4

http://functions.wolfram.com/07.23.03.a9nn.01

Input Form

Hypergeometric2F1[-(19/4), 13/4, 4, -z] == (256 Sqrt[2] (Sqrt[1 + z] (46816 - 71687 z + 237006 z^2 + 4935941 z^3 + 13500224 z^4 + 15717888 z^5 + 8617984 z^6 + 1835008 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (46816 - 24871 z + 165319 z^2 + 5172947 z^3 + 18436165 z^4 + 29218112 z^5 + 24335872 z^6 + 10452992 z^7 + 1835008 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (46816 - 36575 z + 179949 z^2 + 1819751 z^3 + 4238171 z^4 + 4476288 z^5 + 2283520 z^6 + 458752 z^7) EllipticK[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] - Sqrt[1 + z] (46816 - 71687 z + 237006 z^2 + 4935941 z^3 + 13500224 z^4 + 15717888 z^5 + 8617984 z^6 + 1835008 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (422463195 Pi z^3 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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]