Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a95f)

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=-11/4

http://functions.wolfram.com/07.23.03.a95f.01

Input Form

Hypergeometric2F1[-(19/4), -(11/4), 5, -z] == (4096 Sqrt[2] (-4 Sqrt[1 + z] (352 + 6732 z + 73458 z^2 + 784553 z^3 - 13468635 z^4 + 19432638 z^5 - 5327612 z^6 + 93357 z^7 + 2277 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 4 (352 + 7084 z + 80190 z^2 + 858011 z^3 - 12684082 z^4 + 5964003 z^5 + 14105026 z^6 - 5234255 z^7 + 95634 z^8 + 2277 z^9) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (1408 + 27984 z + 313929 z^2 + 3356738 z^3 + 19626195 z^4 - 144362628 z^5 + 129468511 z^6 - 21909294 z^7 + 2277 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + 4 Sqrt[1 + z] (352 + 6732 z + 73458 z^2 + 784553 z^3 - 13468635 z^4 + 19432638 z^5 - 5327612 z^6 + 93357 z^7 + 2277 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (145749802275 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 145749802275 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </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]