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

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-35/8, b>=a

For fixed z and a=-35/8, b=47/8

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

Input Form

Hypergeometric2F1[-(35/8), 47/8, 1, z] == (1/(366134769 Pi)) (2^(3/4) (1 + Sqrt[1 - z])^(1/4) (8 (156226709 - 2806811792 z + 12450174672 z^2 - 19599849984 z^3 + 10016497920 z^4) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 4 (156226709 - 2806811792 z + 12450174672 z^2 - 19599849984 z^3 + 10016497920 z^4) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (1/(Sqrt[1 - z] z)) (3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-122044923 + 3242735906 z - 20610233968 z^2 + 50704111328 z^3 - 53222797056 z^4 + 20032995840 z^5) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]) + (1/z) (Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-366134769 + 9415754300 z - 56217078320 z^2 + 127211984640 z^3 - 120468691200 z^4 + 40065991680 z^5) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])])))

Standard Form

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MathML Form

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<cn type='integer'> 9415754300 </cn> <ci> z </ci> </apply> <cn type='integer'> -366134769 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </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> </apply> </apply> </apply> </apply> </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]