Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8yk)

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=-21/4, b>=a

For fixed z and a=-21/4, b=23/4

http://functions.wolfram.com/07.23.03.a8yk.01

Input Form

Hypergeometric2F1[-(21/4), 23/4, 1/2, z] == (1/(18346020 Pi^(3/2))) ((2 (4586505 - 264798240 z + 2389596160 z^2 - 7420772352 z^3 + 9286189056 z^4 - 4026531840 z^5) EllipticE[(1/2) (1 - Sqrt[z])] + 2 (4586505 - 264798240 z + 2389596160 z^2 - 7420772352 z^3 + 9286189056 z^4 - 4026531840 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (-4586505 + 20784120 Sqrt[z] + 264798240 z - 327416320 z^(3/2) - 2389596160 z^2 + 1355612160 z^(5/2) + 7420772352 z^3 - 2038431744 z^(7/2) - 9286189056 z^4 + 1006632960 z^(9/2) + 4026531840 z^5) EllipticK[(1/2) (1 - Sqrt[z])] + (-4586505 - 20784120 Sqrt[z] + 264798240 z + 327416320 z^(3/2) - 2389596160 z^2 - 1355612160 z^(5/2) + 7420772352 z^3 + 2038431744 z^(7/2) - 9286189056 z^4 - 1006632960 z^(9/2) + 4026531840 z^5) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> 327416320 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 264798240 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20784120 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -4586505 </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'> 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]