Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ahy4)

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=1/4

http://functions.wolfram.com/07.23.03.ahy4.01

Input Form

Hypergeometric2F1[1/4, 1/4, -(11/2), z] == (1/(221760 Pi^(3/2))) ((-((1/(-1 + z)^6) (4 Sqrt[z] (-55440 + 312480 z - 723247 z^2 + 873027 z^3 - 570537 z^4 + 184197 z^5) EllipticE[(1/2) (1 - Sqrt[z])])) + (1/(-1 + z)^6) (4 Sqrt[z] (-55440 + 312480 z - 723247 z^2 + 873027 z^3 - 570537 z^4 + 184197 z^5) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^6 (1 + Sqrt[z])^5)) ((221760 - 332640 Sqrt[z] - 1083600 z + 1708560 z^(3/2) + 2096920 z^2 - 3543414 z^(5/2) - 1990293 z^3 + 3736347 z^(7/2) + 901794 z^4 - 2042868 z^(9/2) - 136341 z^5 + 504735 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^6)) ((-221760 - 332640 Sqrt[z] + 1083600 z + 1708560 z^(3/2) - 2096920 z^2 - 3543414 z^(5/2) + 1990293 z^3 + 3736347 z^(7/2) - 901794 z^4 - 2042868 z^(9/2) + 136341 z^5 + 504735 z^(11/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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/> <cn type='integer'> 332640 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 221760 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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'> 5 </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'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> 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type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3543414 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2096920 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1708560 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1083600 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 332640 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -221760 </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 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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]