Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9969)

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=17/4

http://functions.wolfram.com/07.23.03.9969.01

Input Form

Hypergeometric2F1[-(23/4), 17/4, -(3/2), z] == (1/(91260 Pi^(3/2))) ((4 Sqrt[z] (22815 + 380250 z - 202884096 z^2 + 1162850304 z^3 - 1996488704 z^4 + 1056964608 z^5) EllipticE[(1/2) (1 - Sqrt[z])] - 4 Sqrt[z] (22815 + 380250 z - 202884096 z^2 + 1162850304 z^3 - 1996488704 z^4 + 1056964608 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (91260 - 45630 Sqrt[z] + 1452555 z - 760500 z^(3/2) + 36504000 z^2 + 405768192 z^(5/2) - 381179904 z^3 - 2325700608 z^(7/2) + 849608704 z^4 + 3992977408 z^(9/2) - 528482304 z^5 - 2113929216 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (91260 + 45630 Sqrt[z] + 1452555 z + 760500 z^(3/2) + 36504000 z^2 - 405768192 z^(5/2) - 381179904 z^3 + 2325700608 z^(7/2) + 849608704 z^4 - 3992977408 z^(9/2) - 528482304 z^5 + 2113929216 z^(11/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1452555 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 45630 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 91260 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </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]