Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8vj)

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=19/4

http://functions.wolfram.com/07.23.03.a8vj.01

Input Form

Hypergeometric2F1[-(21/4), 19/4, 7/2, z] == (1/(309806343 Pi^(3/2) z^(5/2))) (2 (-4 Sqrt[z] (3315 + 39780 z - 18502281 z^2 + 122093682 z^3 - 306336768 z^4 + 370470912 z^5 - 218103808 z^6 + 50331648 z^7) EllipticE[(1/2) (1 - Sqrt[z])] - 4 Sqrt[z] (3315 + 39780 z - 18502281 z^2 + 122093682 z^3 - 306336768 z^4 + 370470912 z^5 - 218103808 z^6 + 50331648 z^7) EllipticE[(1/2) (1 + Sqrt[z])] + (-13260 + 6630 Sqrt[z] - 149175 z + 79560 z^(3/2) - 3321630 z^2 - 37004562 z^(5/2) + 37023969 z^3 + 244187364 z^(7/2) - 115137216 z^4 - 612673536 z^(9/2) + 158398464 z^5 + 740941824 z^(11/2) - 101974016 z^6 - 436207616 z^(13/2) + 25165824 z^7 + 100663296 z^(15/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (13260 + 6630 Sqrt[z] + 149175 z + 79560 z^(3/2) + 3321630 z^2 - 37004562 z^(5/2) - 37023969 z^3 + 244187364 z^(7/2) + 115137216 z^4 - 612673536 z^(9/2) - 158398464 z^5 + 740941824 z^(11/2) + 101974016 z^6 - 436207616 z^(13/2) - 25165824 z^7 + 100663296 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='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 101974016 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 740941824 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 158398464 </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'> 612673536 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 115137216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 244187364 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 37023969 </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'> 37004562 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3321630 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 79560 </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'> 149175 </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 /> <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 /> <apply> <plus /> <apply> <times /> <cn type='integer'> 100663296 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 25165824 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 436207616 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 101974016 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> 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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]