Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8e0)

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=-5/4

http://functions.wolfram.com/07.23.03.a8e0.01

Input Form

Hypergeometric2F1[-(21/4), -(5/4), 7/2, z] == (1/(309806343 Pi^(3/2) z^(5/2))) (2 (-4 Sqrt[z] (3315 - 62985 z - 18193986 z^2 - 30295002 z^3 - 2073225 z^4 + 333795 z^5 - 47080 z^6 + 3520 z^7) EllipticE[(1/2) (1 - Sqrt[z])] - 4 Sqrt[z] (3315 - 62985 z - 18193986 z^2 - 30295002 z^3 - 2073225 z^4 + 333795 z^5 - 47080 z^6 + 3520 z^7) EllipticE[(1/2) (1 + Sqrt[z])] + (-13260 + 6630 Sqrt[z] + 261885 z - 125970 z^(3/2) - 4863105 z^2 - 36387972 z^(5/2) - 49606926 z^3 - 60590004 z^(7/2) - 46581150 z^4 - 4146450 z^(9/2) + 160545 z^5 + 667590 z^(11/2) - 23045 z^6 - 94160 z^(13/2) + 1760 z^7 + 7040 z^(15/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (13260 + 6630 Sqrt[z] - 261885 z - 125970 z^(3/2) + 4863105 z^2 - 36387972 z^(5/2) + 49606926 z^3 - 60590004 z^(7/2) + 46581150 z^4 - 4146450 z^(9/2) - 160545 z^5 + 667590 z^(11/2) + 23045 z^6 - 94160 z^(13/2) - 1760 z^7 + 7040 z^(15/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> 60590004 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 49606926 </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'> 36387972 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4863105 </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'> 125970 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 261885 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 6630 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 13260 </cn> </apply> <apply> <ci> EllipticK </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> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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]