Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9975)

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/07.23.03.9975.01

Input Form

Hypergeometric2F1[-(23/4), 17/4, 1, -z] == (1/(19684665 Pi Sqrt[1 + Sqrt[1 + z]])) (2 Sqrt[2] (2 Sqrt[1 + z] (20845177 + 382031194 z + 1876380672 z^2 + 3746541568 z^3 + 3288334336 z^4 + 1056964608 z^5) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + 2 (20845177 + 402876371 z + 2258411866 z^2 + 5622922240 z^3 + 7034875904 z^4 + 4345298944 z^5 + 1056964608 z^6) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (22005689 + 313055861 z + 1294206528 z^2 + 2267568128 z^3 + 1792802816 z^4 + 528482304 z^5) EllipticK[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] - 2 Sqrt[1 + z] (20845177 + 382031194 z + 1876380672 z^2 + 3746541568 z^3 + 3288334336 z^4 + 1056964608 z^5) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))

Standard Form

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MathML Form

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<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> <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='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </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]