Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b5q6)

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=43/8

http://functions.wolfram.com/07.23.03.b5q6.01

Input Form

Hypergeometric2F1[-(47/8), 43/8, 5, z] == (65536 2^(1/4) (-2 Sqrt[1 - z] (1338295296 + 6789060512 z + 35788508093 z^2 + 303224082434 z^3 - 10278345913115 z^4 + 45369094077580 z^5 - 86371595698640 z^6 + 84178919235840 z^7 - 41458284890880 z^8 + 8213528294400 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (1338295296 + 6789060512 z + 35788508093 z^2 + 303224082434 z^3 - 10278345913115 z^4 + 45369094077580 z^5 - 86371595698640 z^6 + 84178919235840 z^7 - 41458284890880 z^8 + 8213528294400 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[1 - z] (1338295296 + 6789060512 z + 35788508093 z^2 + 303224082434 z^3 - 10278345913115 z^4 + 45369094077580 z^5 - 86371595698640 z^6 + 84178919235840 z^7 - 41458284890880 z^8 + 8213528294400 z^9) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 4 (334573824 + 1488156488 z + 7852029432 z^2 + 70023948891 z^3 + 1634741491040 z^4 - 14117896966425 z^5 + 42328431591390 z^6 - 64031740651840 z^7 + 52987409354880 z^8 - 22987862726400 z^9 + 4106764147200 z^10) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (557411929351702425 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^4)

Standard Form

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MathML Form

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<ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1634741491040 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 70023948891 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7852029432 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1488156488 </cn> <ci> z </ci> </apply> <cn type='integer'> 334573824 </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> 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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]