Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ae19)

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=-9/4, b>=a

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

http://functions.wolfram.com/07.23.03.ae19.01

Input Form

Hypergeometric2F1[-(9/4), -(9/4), 11/2, z] == (1/(4367270061 Pi^(3/2) z^(9/2))) (8 (8 Sqrt[z] (-1050 + 13275 z - 85500 z^2 + 422250 z^3 + 31084194 z^4 + 24927171 z^5 + 2359916 z^6) EllipticE[(1/2) (1 - Sqrt[z])] + 8 Sqrt[z] (-1050 + 13275 z - 85500 z^2 + 422250 z^3 + 31084194 z^4 + 24927171 z^5 + 2359916 z^6) EllipticE[(1/2) (1 + Sqrt[z])] - (8400 - 4200 Sqrt[z] - 112500 z + 53100 z^(3/2) + 762705 z^2 - 342000 z^(5/2) - 3879525 z^3 + 1689000 z^(7/2) + 26743050 z^4 + 124336776 z^(9/2) + 136317138 z^5 + 99708684 z^(11/2) + 70499141 z^6 + 9439664 z^(13/2) + 4542615 z^7) EllipticK[(1/2) (1 - Sqrt[z])] + (8400 + 4200 Sqrt[z] - 112500 z - 53100 z^(3/2) + 762705 z^2 + 342000 z^(5/2) - 3879525 z^3 - 1689000 z^(7/2) + 26743050 z^4 - 124336776 z^(9/2) + 136317138 z^5 - 99708684 z^(11/2) + 70499141 z^6 - 9439664 z^(13/2) + 4542615 z^7) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 85500 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 13275 </cn> <ci> z </ci> </apply> <cn type='integer'> -1050 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4542615 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9439664 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 70499141 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> 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z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 762705 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 53100 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 112500 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4200 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 8400 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> 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2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 26743050 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1689000 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3879525 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 342000 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 762705 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 53100 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </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]