Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6hj)

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=-45/8, b>=a

For fixed z and a=-45/8, b=-7/8

http://functions.wolfram.com/07.23.03.b6hj.01

Input Form

Hypergeometric2F1[-(45/8), -(7/8), 6, z] == (524288 2^(1/4) (2 Sqrt[1 - z] (105480192 - 1588383360 z + 12016510905 z^2 - 65314379130 z^3 + 358584641415 z^4 + 2264412821096 z^5 + 590921747815 z^6 - 87561227010 z^7 + 14997345385 z^8 - 1909456780 z^9 + 123856656 z^10) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (105480192 - 1588383360 z + 12016510905 z^2 - 65314379130 z^3 + 358584641415 z^4 + 2264412821096 z^5 + 590921747815 z^6 - 87561227010 z^7 + 14997345385 z^8 - 1909456780 z^9 + 123856656 z^10) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - (105480192 - 1654308480 z + 12998434665 z^2 - 72666897030 z^3 + 398247228015 z^4 - 5763991877764 z^5 - 6899184281425 z^6 - 15297436390 z^7 + 2591573185 z^8 - 324386480 z^9 + 20642776 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[1 - z] (105480192 - 1588383360 z + 12016510905 z^2 - 65314379130 z^3 + 358584641415 z^4 + 2264412821096 z^5 + 590921747815 z^6 - 87561227010 z^7 + 14997345385 z^8 - 1909456780 z^9 + 123856656 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/(2047463474796144405 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^5)

Standard Form

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MathML Form

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<apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 65314379130 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 12016510905 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1588383360 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 105480192 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <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> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn 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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]