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

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=19/4, b>=a

For fixed z and a=19/4, b=23/4

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

Input Form

Hypergeometric2F1[19/4, 23/4, 5/2, z] == (1/(1689765 Pi^(3/2) z^(3/2))) (((1/(-1 + z)^8) (2 Sqrt[z] (585 + 401820 z + 1065846 z^2 + 363292 z^3 + 3465 z^4) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^8) (2 Sqrt[z] (585 + 401820 z + 1065846 z^2 + 363292 z^3 + 3465 z^4) EllipticE[(1/2) (1 + Sqrt[z])]) - (1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7)) ((-1170 + 1755 Sqrt[z] + 40365 z + 361455 z^(3/2) + 264165 z^2 + 801681 z^(5/2) + 151927 z^3 + 211365 z^(7/2) + 3465 z^4) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8)) ((-1170 - 1755 Sqrt[z] + 40365 z - 361455 z^(3/2) + 264165 z^2 - 801681 z^(5/2) + 151927 z^3 - 211365 z^(7/2) + 3465 z^4) 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> <cn type='integer'> -1170 </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> <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]