Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aff7)

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=-7/4, b>=a

For fixed z and a=-7/4, b=21/4

http://functions.wolfram.com/07.23.03.aff7.01

Input Form

Hypergeometric2F1[-(7/4), 21/4, 1/2, z] == (1/(3315 Pi^(3/2))) (((1/(-1 + z)^3) (2 Sqrt[z] (44427 - 318507 z + 702336 z^2 - 624640 z^3 + 196608 z^4) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^3) (2 Sqrt[z] (44427 - 318507 z + 702336 z^2 - 624640 z^3 + 196608 z^4) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^3 (1 + Sqrt[z])^2)) ((-3315 - 41112 Sqrt[z] + 90171 z + 228336 z^(3/2) - 367488 z^2 - 334848 z^(5/2) + 477184 z^3 + 147456 z^(7/2) - 196608 z^4) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z])^3)) ((3315 - 41112 Sqrt[z] - 90171 z + 228336 z^(3/2) + 367488 z^2 - 334848 z^(5/2) - 477184 z^3 + 147456 z^(7/2) + 196608 z^4) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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</apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 90171 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 41112 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 3315 </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'> 3 <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]