Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aag0)

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=-17/4, b>=a

For fixed z and a=-17/4, b=-1/2

http://functions.wolfram.com/07.23.03.aag0.01

Input Form

Hypergeometric2F1[-(17/4), -(1/2), 3, z] == (8 Sqrt[2] (-2 (5304 - 82212 z - 2061765 z^2 - 726400 z^3 + 171200 z^4 - 35392 z^5 + 3696 z^6) EllipticE[1/2 - (1 - z)^(1/4)/ (1 + Sqrt[1 - z])] - 2 Sqrt[1 - z] (5304 - 82212 z - 2061765 z^2 - 726400 z^3 + 171200 z^4 - 35392 z^5 + 3696 z^6) EllipticE[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - (1 - z)^(3/4) (-5304 + 79560 z + 795435 z^2 + 48400 z^3 - 10800 z^4 + 1232 z^5) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + (5304 - 82212 z - 2061765 z^2 - 726400 z^3 + 171200 z^4 - 35392 z^5 + 3696 z^6) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + (1 - z)^(1/4) (5304 - 82212 z - 2061765 z^2 - 726400 z^3 + 171200 z^4 - 35392 z^5 + 3696 z^6) EllipticK[1/2 - (1 - z)^(1/4)/ (1 + Sqrt[1 - z])] + Sqrt[1 - z] (5304 - 82212 z - 2061765 z^2 - 726400 z^3 + 171200 z^4 - 35392 z^5 + 3696 z^6) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])]))/ (5221125 Pi Sqrt[1 + Sqrt[1 - z]] z^2)

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 17 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mn> 3 </mn> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["17", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "2"]]], HypergeometricPFQ, 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<apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3696 </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'> 35392 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 171200 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 726400 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2061765 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 82212 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 5304 </cn> 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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]