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

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=-17/4, b>=a

For fixed z and a=-17/4, b=19/4

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

Input Form

Hypergeometric2F1[-(17/4), 19/4, -(1/2), z] == (1/(20328 Pi^(3/2))) ((-((1/(-1 + z)) (2 (5082 + 200739 z - 3739520 z^2 + 14223360 z^3 - 19070976 z^4 + 8388608 z^5) EllipticE[(1/2) (1 - Sqrt[z])])) - (1/(-1 + z)) (2 (5082 + 200739 z - 3739520 z^2 + 14223360 z^3 - 19070976 z^4 + 8388608 z^5) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/(-1 + Sqrt[z])) ((5082 - 2541 Sqrt[z] + 203280 z - 652160 z^(3/2) - 3087360 z^2 + 5621760 z^(5/2) + 8601600 z^3 - 12779520 z^(7/2) - 6291456 z^4 + 8388608 z^(9/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/(1 + Sqrt[z])) ((-5082 - 2541 Sqrt[z] - 203280 z - 652160 z^(3/2) + 3087360 z^2 + 5621760 z^(5/2) - 8601600 z^3 - 12779520 z^(7/2) + 6291456 z^4 + 8388608 z^(9/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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<cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 652160 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 203280 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2541 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -5082 </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]