Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9fz)

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=5/4

http://functions.wolfram.com/07.23.03.a9fz.01

Input Form

Hypergeometric2F1[-(19/4), 5/4, -(3/2), z] == (1/(180 Pi^(3/2))) ((4 Sqrt[z] (45 + 195 z - 19542 z^2 + 39936 z^3 - 20480 z^4) EllipticE[(1/2) (1 - Sqrt[z])] + 4 Sqrt[z] (-45 - 195 z + 19542 z^2 - 39936 z^3 + 20480 z^4) EllipticE[(1/2) (1 + Sqrt[z])] + (180 - 90 Sqrt[z] + 645 z - 390 z^(3/2) + 5715 z^2 + 39084 z^(5/2) - 17088 z^3 - 79872 z^(7/2) + 10240 z^4 + 40960 z^(9/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (180 + 90 Sqrt[z] + 645 z + 390 z^(3/2) + 5715 z^2 - 39084 z^(5/2) - 17088 z^3 + 79872 z^(7/2) + 10240 z^4 - 40960 z^(9/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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</cn> </apply> </apply> <apply> <times /> <cn type='integer'> 645 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 90 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 180 </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]