Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9v1)

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=-19/4, b>=a

For fixed z and a=-19/4, b=21/4

http://functions.wolfram.com/07.23.03.a9v1.01

Input Form

Hypergeometric2F1[-(19/4), 21/4, 4, -z] == (256 Sqrt[2] (Sqrt[1 + z] (234080 - 2172555 z + 23634765 z^2 + 1208147764 z^3 + 5564325888 z^4 + 9725829120 z^5 + 7457472512 z^6 + 2113929216 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (234080 - 1938475 z + 21462210 z^2 + 1231782529 z^3 + 6772473652 z^4 + 15290155008 z^5 + 17183301632 z^6 + 9571401728 z^7 + 2113929216 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (234080 - 1996995 z + 21988890 z^2 + 496625029 z^3 + 1875702336 z^4 + 2887686144 z^5 + 2013003776 z^6 + 528482304 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (234080 - 2172555 z + 23634765 z^2 + 1208147764 z^3 + 5564325888 z^4 + 9725829120 z^5 + 7457472512 z^6 + 2113929216 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (93364366095 Pi z^3 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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</apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 93364366095 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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]