Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9367)

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=-3/4

http://functions.wolfram.com/07.23.03.9367.01

Input Form

Hypergeometric2F1[-(23/4), -(3/4), -(5/2), z] == (1/(600 Pi^(3/2))) ((2 Sqrt[z] (300 - 405 z - 330 z^2 + 5523 z^3 - 6240 z^4 + 2048 z^5) EllipticE[(1/2) (1 - Sqrt[z])] + 2 Sqrt[z] (-300 + 405 z + 330 z^2 - 5523 z^3 + 6240 z^4 - 2048 z^5) EllipticE[(1/2) (1 + Sqrt[z])] - (-600 + 300 Sqrt[z] + 1260 z - 405 z^(3/2) + 120 z^2 - 330 z^(5/2) + 1020 z^3 + 5523 z^(7/2) - 1416 z^4 - 6240 z^(9/2) + 512 z^5 + 2048 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (600 + 300 Sqrt[z] - 1260 z - 405 z^(3/2) - 120 z^2 - 330 z^(5/2) - 1020 z^3 + 5523 z^(7/2) + 1416 z^4 - 6240 z^(9/2) - 512 z^5 + 2048 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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<apply> <times /> <cn type='integer'> 300 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 600 </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]