Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9ux)

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=21/4

http://functions.wolfram.com/07.23.03.a9ux.01

Input Form

Hypergeometric2F1[-(19/4), 21/4, 3, -z] == (64 Sqrt[2] (2 Sqrt[1 + z] (-14630 + 329175 z + 27393546 z^2 + 171244544 z^3 + 379072512 z^4 + 352321536 z^5 + 117440512 z^6) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + 2 (-14630 + 314545 z + 27722721 z^2 + 198638090 z^3 + 550317056 z^4 + 731394048 z^5 + 469762048 z^6 + 117440512 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (-29260 + 636405 z + 23910477 z^2 + 120788032 z^3 + 231512064 z^4 + 192675840 z^5 + 58720256 z^6) EllipticK[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] - 2 Sqrt[1 + z] (-14630 + 329175 z + 27393546 z^2 + 171244544 z^3 + 379072512 z^4 + 352321536 z^5 + 117440512 z^6) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (1003917915 Pi z^2 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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type='integer'> 1003917915 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </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]