Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bct7)

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=9/8

http://functions.wolfram.com/07.23.03.bct7.01

Input Form

Hypergeometric2F1[-(35/8), 9/8, 5, z] == (65536 2^(1/4) (8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (1444608 - 11120472 z + 36583569 z^2 - 63596610 z^3 + 110747116 z^4 - 95731566 z^5 + 49780395 z^6 - 14580160 z^7 + 1854720 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (5778432 - 46648800 z + 162422469 z^2 - 305015436 z^3 + 125005414 z^4 - 103870668 z^5 + 52348989 z^6 - 14927920 z^7 + 1854720 z^8) EllipticK[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) (1444608 - 11120472 z + 36583569 z^2 - 63596610 z^3 + 110747116 z^4 - 95731566 z^5 + 49780395 z^6 - 14580160 z^7 + 1854720 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 4 Sqrt[1 - z] (1444608 - 11120472 z + 36583569 z^2 - 63596610 z^3 + 110747116 z^4 - 95731566 z^5 + 49780395 z^6 - 14580160 z^7 + 1854720 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(13126535636445 Pi (1 + Sqrt[1 - z])^(1/4) z^4)

Standard Form

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MathML Form

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type='integer'> 11120472 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 1444608 </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> </apply> </apply> </apply> </apply> </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]