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

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=-15/4, b>=a

For fixed z and a=-15/4, b=17/4

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

Input Form

Hypergeometric2F1[-(15/4), 17/4, -(1/2), z] == (1/(1170 Pi^(3/2))) ((-((1/(-1 + z)) (2 Sqrt[z] (585 - 404160 z + 2274816 z^2 - 3702784 z^3 + 1835008 z^4) EllipticE[(1/2) (1 - Sqrt[z])])) + (1/(-1 + z)) (2 Sqrt[z] (585 - 404160 z + 2274816 z^2 - 3702784 z^3 + 1835008 z^4) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/(-1 + Sqrt[z])) ((-1170 + 1755 Sqrt[z] - 37440 z - 366720 z^(3/2) + 744960 z^2 + 1529856 z^(5/2) - 2326528 z^3 - 1376256 z^(7/2) + 1835008 z^4) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/(1 + Sqrt[z])) ((1170 + 1755 Sqrt[z] + 37440 z - 366720 z^(3/2) - 744960 z^2 + 1529856 z^(5/2) + 2326528 z^3 - 1376256 z^(7/2) - 1835008 z^4) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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</apply> </apply> </apply> <apply> <times /> <cn type='integer'> 37440 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 1755 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1170 </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> <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]