Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8j1)

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=-21/4, b>=a

For fixed z and a=-21/4, b=3/4

http://functions.wolfram.com/07.23.03.a8j1.01

Input Form

Hypergeometric2F1[-(21/4), 3/4, 11/2, z] == (1/(6712470765 Pi^(3/2) z^(9/2))) (8 (-8 Sqrt[z] (46410 - 467415 z + 2162043 z^2 - 6260709 z^3 - 36816780 z^4 + 26141115 z^5 - 14106785 z^6 + 5155689 z^7 - 1132032 z^8 + 112640 z^9) EllipticE[(1/2) (1 - Sqrt[z])] - 8 Sqrt[z] (46410 - 467415 z + 2162043 z^2 - 6260709 z^3 - 36816780 z^4 + 26141115 z^5 - 14106785 z^6 + 5155689 z^7 - 1132032 z^8 + 112640 z^9) EllipticE[(1/2) (1 + Sqrt[z])] + (-371280 + 185640 Sqrt[z] + 4017780 z - 1869660 z^(3/2) - 20059065 z^2 + 8648172 z^(5/2) + 62658141 z^3 - 25042836 z^(7/2) - 160810650 z^4 - 147267120 z^(9/2) + 22852830 z^5 + 104564460 z^(11/2) - 12817805 z^6 - 56427140 z^(13/2) + 4854465 z^7 + 20622756 z^(15/2) - 1100352 z^8 - 4528128 z^(17/2) + 112640 z^9 + 450560 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (371280 + 185640 Sqrt[z] - 4017780 z - 1869660 z^(3/2) + 20059065 z^2 + 8648172 z^(5/2) - 62658141 z^3 - 25042836 z^(7/2) + 160810650 z^4 - 147267120 z^(9/2) - 22852830 z^5 + 104564460 z^(11/2) + 12817805 z^6 - 56427140 z^(13/2) - 4854465 z^7 + 20622756 z^(15/2) + 1100352 z^8 - 4528128 z^(17/2) - 112640 z^9 + 450560 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 112640 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1132032 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5155689 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 14106785 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 26141115 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 36816780 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6260709 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2162043 </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'> 467415 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 46410 </cn> </apply> <apply> <ci> EllipticE </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> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 112640 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type='integer'> 8648172 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20059065 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1869660 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4017780 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 185640 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -371280 </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 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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]