Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9al)

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=-3/4

http://functions.wolfram.com/07.23.03.a9al.01

Input Form

Hypergeometric2F1[-(19/4), -(3/4), 6, -z] == (16384 Sqrt[2] ((-Sqrt[1 + z]) (2048 + 27552 z + 183571 z^2 + 862883 z^3 + 4003566 z^4 - 18313498 z^5 + 2457343 z^6 + 348519 z^7 + 43792 z^8 + 2944 z^9) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (2048 + 29600 z + 211123 z^2 + 1046454 z^3 + 4866449 z^4 - 14309932 z^5 - 15856155 z^6 + 2805862 z^7 + 392311 z^8 + 46736 z^9 + 2944 z^10) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (2048 + 29088 z + 204091 z^2 + 998690 z^3 + 4638669 z^4 + 20612732 z^5 - 30648443 z^6 + 90114 z^7 + 11155 z^8 + 736 z^9) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + Sqrt[1 + z] (2048 + 27552 z + 183571 z^2 + 862883 z^3 + 4003566 z^4 - 18313498 z^5 + 2457343 z^6 + 348519 z^7 + 43792 z^8 + 2944 z^9) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (294738489045 Pi z^5 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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<apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 998690 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 204091 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 29088 </cn> <ci> z </ci> </apply> <cn type='integer'> 2048 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <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> <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='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus 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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]