Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9719)

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=9/4

http://functions.wolfram.com/07.23.03.9719.01

Input Form

Hypergeometric2F1[-(23/4), 9/4, 3/2, z] == (1/(580125 Pi^(3/2) Sqrt[z])) (2 (2 (100947 - 2050566 z + 9161075 z^2 - 17967040 z^3 + 17999360 z^4 - 9076736 z^5 + 1835008 z^6) EllipticE[(1/2) (1 - Sqrt[z])] - 2 (100947 - 2050566 z + 9161075 z^2 - 17967040 z^3 + 17999360 z^4 - 9076736 z^5 + 1835008 z^6) EllipticE[(1/2) (1 + Sqrt[z])] - (100947 - 239589 Sqrt[z] - 2050566 z + 1516970 z^(3/2) + 9161075 z^2 - 3525445 z^(5/2) - 17967040 z^3 + 3931520 z^(7/2) + 17999360 z^4 - 2140160 z^(9/2) - 9076736 z^5 + 458752 z^(11/2) + 1835008 z^6) EllipticK[(1/2) (1 - Sqrt[z])] + (100947 + 239589 Sqrt[z] - 2050566 z - 1516970 z^(3/2) + 9161075 z^2 + 3525445 z^(5/2) - 17967040 z^3 - 3931520 z^(7/2) + 17999360 z^4 + 2140160 z^(9/2) - 9076736 z^5 - 458752 z^(11/2) + 1835008 z^6) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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/> <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 /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1835008 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 458752 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9076736 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2140160 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 17999360 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type='integer'> 2050566 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 239589 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 100947 </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]