Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9109)

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=-15/4

http://functions.wolfram.com/07.23.03.9109.01

Input Form

Hypergeometric2F1[-(23/4), -(15/4), 1/2, z] == (1/(129285 Pi^(3/2))) ((8 Sqrt[z] (-677850 - 7278135 z - 12830368 z^2 - 4317266 z^3 - 64350 z^4 + 2145 z^5) EllipticE[(1/2) (1 - Sqrt[z])] - 8 Sqrt[z] (-677850 - 7278135 z - 12830368 z^2 - 4317266 z^3 - 64350 z^4 + 2145 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (129285 + 2711400 Sqrt[z] + 6914955 z + 29112540 z^(3/2) + 38242210 z^2 + 51321472 z^(5/2) + 44857766 z^3 + 17269064 z^(7/2) + 10521225 z^4 + 257400 z^(9/2) - 2145 z^5 - 8580 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (129285 - 2711400 Sqrt[z] + 6914955 z - 29112540 z^(3/2) + 38242210 z^2 - 51321472 z^(5/2) + 44857766 z^3 - 17269064 z^(7/2) + 10521225 z^4 - 257400 z^(9/2) - 2145 z^5 + 8580 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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<cn type='integer'> 6914955 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2711400 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 129285 </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]