Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ad2r)

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=-3/4

http://functions.wolfram.com/07.23.03.ad2r.01

Input Form

Hypergeometric2F1[-(11/4), -(3/4), -(11/2), z] == (1/(960 Pi^(3/2))) ((-((4 Sqrt[z] (-240 + 480 z - 257 z^2 + 17 z^3 + 2 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^2) + (4 Sqrt[z] (-240 + 480 z - 257 z^2 + 17 z^3 + 2 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^2 + (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z]))) ((960 - 1440 Sqrt[z] - 1200 z + 2160 z^(3/2) + 200 z^2 - 714 z^(5/2) + 37 z^3 - 3 z^(7/2) + 4 z^4) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/(-1 - Sqrt[z] + z + z^(3/2))) ((960 + 1440 Sqrt[z] - 1200 z - 2160 z^(3/2) + 200 z^2 + 714 z^(5/2) + 37 z^3 + 3 z^(7/2) + 4 z^4) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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<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> </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

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["11", "4"]]], ",", RowBox[List["-", FractionBox["3", "4"]]], ",", RowBox[List["-", FractionBox["11", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "240"]], "+", RowBox[List["480", " ", "z"]], "-", RowBox[List["257", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["17", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]]]], "+", FractionBox[RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "240"]], "+", RowBox[List["480", " ", "z"]], "-", RowBox[List["257", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["17", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["960", "-", RowBox[List["1440", " ", SqrtBox["z"]]], "-", RowBox[List["1200", " ", "z"]], "+", RowBox[List["2160", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["200", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["714", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["37", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["3", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["4", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["960", "+", RowBox[List["1440", " ", SqrtBox["z"]]], "-", RowBox[List["1200", " ", "z"]], "-", RowBox[List["2160", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["200", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["714", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["37", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["3", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["4", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "-", SqrtBox["z"], "+", "z", "+", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["960", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]

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]