Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aj77)

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=3/4, b>=a

For fixed z and a=3/4, b=19/4

http://functions.wolfram.com/07.23.03.aj77.01

Input Form

Hypergeometric2F1[3/4, 19/4, 7/2, z] == (1/(1617 Pi^(3/2) z^(5/2))) (2 (-((4 Sqrt[z] (5 + 5 z - 88 z^2 + 64 z^3) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^2) - (4 Sqrt[z] (5 + 5 z - 88 z^2 + 64 z^3) EllipticE[(1/2) (1 + Sqrt[z])])/ (-1 + z)^2 + ((-20 + 30 Sqrt[z] - 35 z + 45 z^(3/2) - 80 z^2 - 96 z^(5/2) + 128 z^3) EllipticK[(1/2) (1 - Sqrt[z])])/ ((-1 + Sqrt[z])^2 (1 + Sqrt[z])) + ((-20 - 30 Sqrt[z] - 35 z - 45 z^(3/2) - 80 z^2 + 96 z^(5/2) + 128 z^3) EllipticK[(1/2) (1 + Sqrt[z])])/(-1 - Sqrt[z] + z + z^(3/2))) Gamma[1/4]^2)

Standard Form

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MathML Form

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/> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </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[FractionBox["3", "4"], ",", FractionBox["19", "4"], ",", FractionBox["7", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["5", " ", "z"]], "-", RowBox[List["88", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["64", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", 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["5", "+", RowBox[List["5", " ", "z"]], "-", RowBox[List["88", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["64", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", 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[RowBox[List["-", "20"]], "+", RowBox[List["30", " ", SqrtBox["z"]]], "-", RowBox[List["35", " ", "z"]], "+", RowBox[List["45", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["80", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["96", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["128", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", 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[RowBox[List["-", "20"]], "-", RowBox[List["30", " ", SqrtBox["z"]]], "-", RowBox[List["35", " ", "z"]], "-", RowBox[List["45", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["80", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["96", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["128", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", 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["1", "4"], "]"]], "2"]]], RowBox[List["1617", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["5", "/", "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]