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

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=-17/4, b>=a

For fixed z and a=-17/4, b=11/4

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

Input Form

Hypergeometric2F1[-(17/4), 11/4, 7/2, z] == (1/(6359661 Pi^(3/2) z^(5/2))) (2 (2 Sqrt[z] (-6630 - 9945 z + 737028 z^2 - 2196165 z^3 + 2667648 z^4 - 1505280 z^5 + 327680 z^6) EllipticE[(1/2) (1 - Sqrt[z])] + 2 Sqrt[z] (-6630 - 9945 z + 737028 z^2 - 2196165 z^3 + 2667648 z^4 - 1505280 z^5 + 327680 z^6) EllipticE[(1/2) (1 + Sqrt[z])] - (13260 - 6630 Sqrt[z] + 9945 z - 9945 z^(3/2) + 99450 z^2 + 737028 z^(5/2) - 410511 z^3 - 2196165 z^(7/2) + 573552 z^4 + 2667648 z^(9/2) - 353280 z^5 - 1505280 z^(11/2) + 81920 z^6 + 327680 z^(13/2)) EllipticK[(1/2) (1 - Sqrt[z])] - (-13260 - 6630 Sqrt[z] - 9945 z - 9945 z^(3/2) - 99450 z^2 + 737028 z^(5/2) + 410511 z^3 - 2196165 z^(7/2) - 573552 z^4 + 2667648 z^(9/2) + 353280 z^5 - 1505280 z^(11/2) - 81920 z^6 + 327680 z^(13/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9945 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9945 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6630 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 327680 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 81920 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1505280 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 353280 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2667648 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 573552 </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'> 2196165 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<cn type='rational'> 1 <sep /> 2 </cn> </apply> </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> <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]