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

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=-9/4, b>=a

For fixed z and a=-9/4, b=19/4

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

Input Form

Hypergeometric2F1[-(9/4), 19/4, 1/2, z] == (1/(6468 Pi^(3/2))) (((2 (1617 - 35109 z + 129664 z^2 - 161792 z^3 + 65536 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^2 + (2 (1617 - 35109 z + 129664 z^2 - 161792 z^3 + 65536 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^2 + (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z]))) ((-1617 + 5781 Sqrt[z] + 29328 z - 52864 z^(3/2) - 76800 z^2 + 112640 z^(5/2) + 49152 z^3 - 65536 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z]) (1 + Sqrt[z])^2)) ((1617 + 5781 Sqrt[z] - 29328 z - 52864 z^(3/2) + 76800 z^2 + 112640 z^(5/2) - 49152 z^3 - 65536 z^(7/2)) EllipticK[(1/2) (1 + 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'> 29328 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 5781 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1617 </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]