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

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=4, b>=a

For fixed z and a=4, b=5

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

Input Form

Hypergeometric2F1[4, 5, -(15/4), z] == (1/69206016) ((1/(-1 + z)^12) (8 (8650752 - 149946368 z + 1376256000 z^2 - 9981394944 z^3 + 92781936640 z^4 + 592113432615 z^5 + 483572294172 z^6 + 73027715296 z^7 + 515002752 z^8)) - (1/(1 - z)^(51/4)) (281642130 Sqrt[2] z^(19/4) (6417 + 13392 z + 5952 z^2 + 512 z^3) ArcTan[1 - z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))]) - (1/(1 - z)^(51/4)) (281642130 Sqrt[2] z^(19/4) (6417 + 13392 z + 5952 z^2 + 512 z^3) ArcTan[1 + z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))]) - (140821065 Sqrt[2] z^(19/4) (6417 + 13392 z + 5952 z^2 + 512 z^3) Log[1 - (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]])/ (1 - z)^(51/4) + (140821065 Sqrt[2] z^(19/4) (6417 + 13392 z + 5952 z^2 + 512 z^3) Log[1 + (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]])/(1 - z)^(51/4))

Standard Form

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MathML Form

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type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <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> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 19 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 51 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> 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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]