Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9rv)

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=-19/4, b>=a

For fixed z and a=-19/4, b=9/2

http://functions.wolfram.com/07.23.03.a9rv.01

Input Form

Hypergeometric2F1[-(19/4), 9/2, 2, z] == (2 (1 - z)^(1/4) (-5016 + 512744 z - 3753090 z^2 + 9147411 z^3 - 9075144 z^4 + 3172455 z^5) EllipticE[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + 2 (1 - z)^(3/4) (-5016 + 512744 z - 3753090 z^2 + 9147411 z^3 - 9075144 z^4 + 3172455 z^5) EllipticE[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (1 - z)^(1/4) (5016 - 512744 z + 3753090 z^2 - 9147411 z^3 + 9075144 z^4 - 3172455 z^5) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + Sqrt[1 - z] (5016 - 512744 z + 3753090 z^2 - 9147411 z^3 + 9075144 z^4 - 3172455 z^5) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (1 - z)^(3/4) (5016 - 512744 z + 3753090 z^2 - 9147411 z^3 + 9075144 z^4 - 3172455 z^5) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (5016 - 111464 z - 356966 z^2 + 4380441 z^3 - 10454847 z^4 + 9709635 z^5 - 3172455 z^6) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)])/ (100947 Sqrt[2] Pi Sqrt[1 + Sqrt[1 - z]] z)

Standard Form

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MathML Form

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type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3753090 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 512744 </cn> <ci> z </ci> </apply> <cn type='integer'> -5016 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <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> <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> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </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]