Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.adl3)

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=-11/4, b>=a

For fixed z and a=-11/4, b=17/4

http://functions.wolfram.com/07.23.03.adl3.01

Input Form

Hypergeometric2F1[-(11/4), 17/4, 5, -z] == (4096 Sqrt[2] (Sqrt[1 + z] (-9856 + 11088 z - 17325 z^2 + 47047 z^3 + 590976 z^4 + 829440 z^5 + 327680 z^6) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (-9856 + 1232 z - 6237 z^2 + 29722 z^3 + 638023 z^4 + 1420416 z^5 + 1157120 z^6 + 327680 z^7) EllipticE[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] - 4 (-2464 + 924 z - 2079 z^2 + 8239 z^3 + 48876 z^4 + 57600 z^5 + 20480 z^6) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (-9856 + 11088 z - 17325 z^2 + 47047 z^3 + 590976 z^4 + 829440 z^5 + 327680 z^6) EllipticK[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])]))/(885809925 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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<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 <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]