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

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=7/4

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

Input Form

Hypergeometric2F1[-(1/4), 7/4, -(5/2), z] == (1/(480 Pi^(3/2))) (((8 (30 - 111 z + 143 z^2 - 45 z^3 + 15 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^4 + (8 (30 - 111 z + 143 z^2 - 45 z^3 + 15 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^4 + (1/((-1 + Sqrt[z])^4 (1 + Sqrt[z])^3)) ((-120 + 60 Sqrt[z] + 384 z - 167 z^(3/2) - 405 z^2 + 135 z^(5/2) + 45 z^3 - 60 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + ((120 + 60 Sqrt[z] - 384 z - 167 z^(3/2) + 405 z^2 + 135 z^(5/2) - 45 z^3 - 60 z^(7/2)) EllipticK[(1/2) (1 + Sqrt[z])])/ ((-1 + Sqrt[z])^3 (1 + Sqrt[z])^4)) Gamma[1/4]^2)

Standard Form

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MathML Form

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<apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </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'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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]