Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9cz)

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=1/4

http://functions.wolfram.com/07.23.03.a9cz.01

Input Form

Hypergeometric2F1[-(19/4), 1/4, 4, -z] == (256 Sqrt[2] (Sqrt[1 + z] (-3040 - 30685 z - 174420 z^2 + 345682 z^3 + 172204 z^4 + 69795 z^5 + 17696 z^6 + 2048 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (-3040 - 33725 z - 205105 z^2 + 171262 z^3 + 517886 z^4 + 241999 z^5 + 87491 z^6 + 19744 z^7 + 2048 z^8) EllipticE[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] - (-3040 - 32965 z - 197220 z^2 - 883238 z^3 + 47428 z^4 + 18627 z^5 + 4568 z^6 + 512 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (-3040 - 30685 z - 174420 z^2 + 345682 z^3 + 172204 z^4 + 69795 z^5 + 17696 z^6 + 2048 z^7) EllipticK[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])]))/(140821065 Pi z^3 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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</apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 140821065 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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]