Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9279)

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=-23/4, b>=a

For fixed z and a=-23/4, b=-7/4

http://functions.wolfram.com/07.23.03.9279.01

Input Form

Hypergeometric2F1[-(23/4), -(7/4), -(3/2), z] == (1/(2340 Pi^(3/2))) ((4 Sqrt[z] (585 - 3705 z - 31521 z^2 + 20937 z^3 - 8120 z^4 + 1344 z^5) EllipticE[(1/2) (1 - Sqrt[z])] + 4 Sqrt[z] (-585 + 3705 z + 31521 z^2 - 20937 z^3 + 8120 z^4 - 1344 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (2340 - 1170 Sqrt[z] - 16575 z + 7410 z^(3/2) + 61425 z^2 + 63042 z^(5/2) - 9429 z^3 - 41874 z^(7/2) + 3871 z^4 + 16240 z^(9/2) - 672 z^5 - 2688 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (2340 + 1170 Sqrt[z] - 16575 z - 7410 z^(3/2) + 61425 z^2 - 63042 z^(5/2) - 9429 z^3 + 41874 z^(7/2) + 3871 z^4 - 16240 z^(9/2) - 672 z^5 + 2688 z^(11/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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</apply> </apply> <apply> <times /> <cn type='integer'> 1170 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 2340 </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'> 3 <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]