Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ai1y)

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=1/4, b>=a

For fixed z and a=1/4, b=5/4

http://functions.wolfram.com/07.23.03.ai1y.01

Input Form

Hypergeometric2F1[1/4, 5/4, -(11/2), z] == (1/(73920 Pi^(3/2))) (((1/(-1 + z)^7) (2 Sqrt[z] (-36960 + 246960 z - 702198 z^2 + 1099725 z^3 - 1024611 z^4 + 576519 z^5 + 168245 z^6) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^7) (2 Sqrt[z] (-36960 + 246960 z - 702198 z^2 + 1099725 z^3 - 1024611 z^4 + 576519 z^5 + 168245 z^6) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^6)) ((-73920 + 110880 Sqrt[z] + 438480 z - 685440 z^(3/2) - 1081080 z^2 + 1783278 z^(5/2) + 1418061 z^3 - 2517786 z^(7/2) - 1048167 z^4 + 2072778 z^(9/2) + 432951 z^5 - 1009470 z^(11/2) - 168245 z^6) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^6 (1 + Sqrt[z])^7)) ((73920 + 110880 Sqrt[z] - 438480 z - 685440 z^(3/2) + 1081080 z^2 + 1783278 z^(5/2) - 1418061 z^3 - 2517786 z^(7/2) + 1048167 z^4 + 2072778 z^(9/2) - 432951 z^5 - 1009470 z^(11/2) + 168245 z^6) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 702198 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 246960 </cn> <ci> z </ci> </apply> <cn type='integer'> -36960 </cn> </apply> <apply> <ci> EllipticE </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> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <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'> 7 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> 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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]