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

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=-5/4, b>=a

For fixed z and a=-5/4, b=-1/4

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

Input Form

Hypergeometric2F1[-(5/4), -(1/4), -(9/2), z] == (1/(24192 Pi^(3/2))) ((-((2 (6048 - 17808 z + 17466 z^2 - 5791 z^3 + 117 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^3) - (2 (6048 - 17808 z + 17466 z^2 - 5791 z^3 + 117 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^3 + (1/((-1 + Sqrt[z])^3 (1 + Sqrt[z])^2)) ((6048 - 3024 Sqrt[z] - 14784 z + 6132 z^(3/2) + 11334 z^2 - 3217 z^(5/2) - 2574 z^3 + 117 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z])^3)) ((-6048 - 3024 Sqrt[z] + 14784 z + 6132 z^(3/2) - 11334 z^2 - 3217 z^(5/2) + 2574 z^3 + 117 z^(7/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <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["5", "4"]]], ",", RowBox[List["-", FractionBox["1", "4"]]], ",", RowBox[List["-", FractionBox["9", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["6048", "-", RowBox[List["17808", " ", "z"]], "+", RowBox[List["17466", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["5791", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["117", " ", 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"]], ")"]], "3"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["6048", "-", RowBox[List["17808", " ", "z"]], "+", RowBox[List["17466", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["5791", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["117", " ", 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"]], ")"]], "3"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["6048", "-", RowBox[List["3024", " ", SqrtBox["z"]]], "-", RowBox[List["14784", " ", "z"]], "+", RowBox[List["6132", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["11334", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["3217", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["2574", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["117", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]]]], ")"]], " ", 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"]]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "6048"]], "-", RowBox[List["3024", " ", SqrtBox["z"]]], "+", RowBox[List["14784", " ", "z"]], "+", RowBox[List["6132", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["11334", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["3217", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["2574", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["117", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]]]], ")"]], " ", 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"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "3"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], "2"]]], RowBox[List["24192", " ", 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]