Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.amw2)

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=23/4

http://functions.wolfram.com/07.23.03.amw2.01

Input Form

Hypergeometric2F1[15/4, 23/4, 3/2, z] == (1/(675906 Pi^(3/2))) ((-((8 (-41001 - 241644 z - 171718 z^2 - 4620 z^3 + 231 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^8) - (8 (-41001 - 241644 z - 171718 z^2 - 4620 z^3 + 231 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^8 + ((-9945 - 154059 Sqrt[z] - 186513 z - 780063 z^(3/2) - 245431 z^2 - 441441 z^(5/2) - 17787 z^3 - 693 z^(7/2) + 924 z^4) EllipticK[(1/2) (1 - Sqrt[z])])/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7 Sqrt[z]) + ((-9945 + 154059 Sqrt[z] - 186513 z + 780063 z^(3/2) - 245431 z^2 + 441441 z^(5/2) - 17787 z^3 + 693 z^(7/2) + 924 z^4) EllipticK[(1/2) (1 + Sqrt[z])])/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8 Sqrt[z])) Gamma[1/4]^2)

Standard Form

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MathML Form

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2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 186513 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 154059 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -9945 </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'> 7 </cn> </apply> 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<times /> <cn type='integer'> 245431 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 780063 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 186513 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 154059 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -9945 </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> <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]