Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b5mh)

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=-47/8, b>=a

For fixed z and a=-47/8, b=37/8

http://functions.wolfram.com/07.23.03.b5mh.01

Input Form

Hypergeometric2F1[-(47/8), 37/8, 3, z] == (256 2^(1/4) (-8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (871286 + 18297006 z - 898295391 z^2 + 5414920357 z^3 - 13159194048 z^4 + 15663884544 z^5 - 9136275456 z^6 + 2095644672 z^7) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (871286 + 18297006 z - 898295391 z^2 + 5414920357 z^3 - 13159194048 z^4 + 15663884544 z^5 - 9136275456 z^6 + 2095644672 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[1 - z] (871286 + 18297006 z - 898295391 z^2 + 5414920357 z^3 - 13159194048 z^4 + 15663884544 z^5 - 9136275456 z^6 + 2095644672 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (3485144 + 71881095 z + 3560947782 z^2 - 42610513253 z^3 + 167444711280 z^4 - 316854058752 z^5 + 317101424640 z^6 - 162247016448 z^7 + 33530314752 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (919287323955 Pi (1 + Sqrt[1 - z])^(1/4) z^2)

Standard Form

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MathML Form

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type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 71881095 </cn> <ci> z </ci> </apply> <cn type='integer'> 3485144 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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 /> 4 </cn> </apply> <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> 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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]