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

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=19/4

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

Input Form

Hypergeometric2F1[15/4, 19/4, 9/2, z] == (1/(2541 Pi^(3/2) z^(7/2))) (4 ((2 Sqrt[z] (20 - 75 z + 106 z^2 + 77 z^3) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^4 + (2 Sqrt[z] (20 - 75 z + 106 z^2 + 77 z^3) EllipticE[(1/2) (1 + Sqrt[z])])/ (-1 + z)^4 - ((-40 + 60 Sqrt[z] + 120 z - 195 z^(3/2) - 125 z^2 + 231 z^(5/2) + 77 z^3) EllipticK[(1/2) (1 - Sqrt[z])])/ ((-1 + Sqrt[z])^4 (1 + Sqrt[z])^3) - ((-40 - 60 Sqrt[z] + 120 z + 195 z^(3/2) - 125 z^2 - 231 z^(5/2) + 77 z^3) EllipticK[(1/2) (1 + Sqrt[z])])/((-1 + Sqrt[z])^3 (1 + Sqrt[z])^4)) Gamma[1/4]^2)

Standard Form

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MathML Form

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