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

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=21/4

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

Input Form

Hypergeometric2F1[21/4, 21/4, 5/2, z] == (1/(10989225 Pi^(3/2) z^(3/2))) (4 (-((1/(-1 + z)^8) (4 (-17787 + 640332 z + 6179326 z^2 + 6189580 z^3 + 771109 z^4) EllipticE[(1/2) (1 - Sqrt[z])])) + (1/(-1 + z)^8) (4 (-17787 + 640332 z + 6179326 z^2 + 6189580 z^3 + 771109 z^4) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7)) ((-35574 + 17787 Sqrt[z] + 1262877 z + 2123279 z^(3/2) + 10235373 z^2 + 3963761 z^(5/2) + 8415399 z^3 + 776453 z^(7/2) + 765765 z^4) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8)) ((-35574 - 17787 Sqrt[z] + 1262877 z - 2123279 z^(3/2) + 10235373 z^2 - 3963761 z^(5/2) + 8415399 z^3 - 776453 z^(7/2) + 765765 z^4) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^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> </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]