Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9105)

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=-23/4, b>=a

For fixed z and a=-23/4, b=-15/4

http://functions.wolfram.com/07.23.03.9105.01

Input Form

Hypergeometric2F1[-(23/4), -(15/4), -(3/2), z] == (1/(780 Pi^(3/2))) ((4 Sqrt[z] (195 - 2730 z - 450198 z^2 - 451296 z^3 - 14245 z^4 + 770 z^5) EllipticE[(1/2) (1 - Sqrt[z])] + 4 Sqrt[z] (-195 + 2730 z + 450198 z^2 + 451296 z^3 + 14245 z^4 - 770 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (780 - 390 Sqrt[z] - 11505 z + 5460 z^(3/2) + 150540 z^2 + 900396 z^(5/2) + 1073418 z^3 + 902592 z^(7/2) + 622160 z^4 + 28490 z^(9/2) - 385 z^5 - 1540 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (780 + 390 Sqrt[z] - 11505 z - 5460 z^(3/2) + 150540 z^2 - 900396 z^(5/2) + 1073418 z^3 - 902592 z^(7/2) + 622160 z^4 - 28490 z^(9/2) - 385 z^5 + 1540 z^(11/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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390 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 780 </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> <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]