Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a92z)

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=-15/4

http://functions.wolfram.com/07.23.03.a92z.01

Input Form

Hypergeometric2F1[-(19/4), -(15/4), 5, -z] == (4096 Sqrt[2] (Sqrt[1 + z] (-4224 - 93104 z - 1196811 z^2 - 15492246 z^3 + 361275143 z^4 - 760931028 z^5 + 382930611 z^6 - 41764742 z^7 + 211761 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (-4224 - 97328 z - 1289915 z^2 - 16689057 z^3 + 345782897 z^4 - 399655885 z^5 - 378000417 z^6 + 341165869 z^7 - 41552981 z^8 + 211761 z^9) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (-4224 - 93104 z - 1196811 z^2 - 15492246 z^3 + 361275143 z^4 - 760931028 z^5 + 382930611 z^6 - 41764742 z^7 + 211761 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + 2 (2112 + 48136 z + 633171 z^2 + 8191722 z^3 + 45748781 z^4 - 527699724 z^5 + 728602557 z^6 - 249474950 z^7 + 17576163 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (903648774105 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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<power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 48136 </cn> <ci> z </ci> </apply> <cn type='integer'> 2112 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <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> <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='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 903648774105 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 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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]