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

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=-33/8

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

Input Form

Hypergeometric2F1[-(35/8), -(33/8), 5, z] == (1/(10646761839038325 Pi z^4)) (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (16 (-380160 + 8605080 z - 112879305 z^2 + 1469208510 z^3 + 123721336649 z^4 + 386772842844 z^5 + 300830576409 z^6 + 62758753134 z^7 + 2419764039 z^8) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (190080 - 4164435 z + 53427330 z^2 + 107608389075 z^3 + 377624095388 z^4 + 318312505731 z^5 + 71308668354 z^6 + 2964715677 z^7) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-190080 + 4271355 z - 55748385 z^2 + 65708343855 z^3 + 260986493107 z^4 + 273831320481 z^5 + 92462158701 z^6 + 9200630901 z^7 + 156981825 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 8 (-380160 + 8605080 z - 112879305 z^2 + 1469208510 z^3 + 123721336649 z^4 + 386772842844 z^5 + 300830576409 z^6 + 62758753134 z^7 + 2419764039 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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<ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 273831320481 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 260986493107 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 65708343855 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 55748385 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4271355 </cn> <ci> z </ci> </apply> <cn type='integer'> -190080 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2419764039 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 62758753134 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 300830576409 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </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]