Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a901)

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=-19/4

http://functions.wolfram.com/07.23.03.a901.01

Input Form

Hypergeometric2F1[-(19/4), -(19/4), -(1/2), z] == (1/(15210 Pi^(3/2))) ((2 Sqrt[z] (7605 + 7686060 z + 29961918 z^2 + 19289260 z^3 + 1775413 z^4) EllipticE[(1/2) (1 - Sqrt[z])] - 2 Sqrt[z] (7605 + 7686060 z + 29961918 z^2 + 19289260 z^3 + 1775413 z^4) EllipticE[(1/2) (1 + Sqrt[z])] - (-15210 + 7605 Sqrt[z] + 692055 z + 7686060 z^(3/2) + 13410120 z^2 + 29961918 z^(5/2) + 30281794 z^3 + 19289260 z^(7/2) + 13495642 z^4 + 1775413 z^(9/2) + 855855 z^5) EllipticK[(1/2) (1 - Sqrt[z])] + (15210 + 7605 Sqrt[z] - 692055 z + 7686060 z^(3/2) - 13410120 z^2 + 29961918 z^(5/2) - 30281794 z^3 + 19289260 z^(7/2) - 13495642 z^4 + 1775413 z^(9/2) - 855855 z^5) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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</cn> </apply> <cn type='integer'> 1 </cn> </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["19", "4"]]], ",", RowBox[List["-", FractionBox["19", "4"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["7605", "+", RowBox[List["7686060", " ", "z"]], "+", RowBox[List["29961918", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["19289260", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1775413", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["7605", "+", RowBox[List["7686060", " ", "z"]], "+", RowBox[List["29961918", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["19289260", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1775413", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "15210"]], "+", RowBox[List["7605", " ", SqrtBox["z"]]], "+", RowBox[List["692055", " ", "z"]], "+", RowBox[List["7686060", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["13410120", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["29961918", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["30281794", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["19289260", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["13495642", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1775413", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["855855", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["15210", "+", RowBox[List["7605", " ", SqrtBox["z"]]], "-", RowBox[List["692055", " ", "z"]], "+", RowBox[List["7686060", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["13410120", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["29961918", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["30281794", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["19289260", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["13495642", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1775413", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "-", RowBox[List["855855", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["15210", " ", 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]