Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9841)

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=13/4

http://functions.wolfram.com/07.23.03.9841.01

Input Form

Hypergeometric2F1[-(23/4), 13/4, -(1/2), z] == (1/(19890 Pi^(3/2))) ((2 Sqrt[z] (9945 - 9340725 z + 69881472 z^2 - 170178560 z^3 + 168361984 z^4 - 58720256 z^5) EllipticE[(1/2) (1 - Sqrt[z])] + 2 Sqrt[z] (-9945 + 9340725 z - 69881472 z^2 + 170178560 z^3 - 168361984 z^4 + 58720256 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (19890 - 9945 Sqrt[z] + 735930 z + 9340725 z^(3/2) - 10431120 z^2 - 69881472 z^(5/2) + 32943104 z^3 + 170178560 z^(7/2) - 37961728 z^4 - 168361984 z^(9/2) + 14680064 z^5 + 58720256 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (19890 + 9945 Sqrt[z] + 735930 z - 9340725 z^(3/2) - 10431120 z^2 + 69881472 z^(5/2) + 32943104 z^3 - 170178560 z^(7/2) - 37961728 z^4 + 168361984 z^(9/2) + 14680064 z^5 - 58720256 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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type='integer'> 735930 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 9945 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 19890 </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]