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

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.andp.01

Input Form

Hypergeometric2F1[19/4, 19/4, 9/2, z] == (1/(38115 Pi^(3/2) z^(7/2))) (4 (-((2 Sqrt[z] (20 - 105 z + 246 z^2 + 1887 z^3) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^5) - (2 Sqrt[z] (20 - 105 z + 246 z^2 + 1887 z^3) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^5 + (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^4)) ((-40 + 60 Sqrt[z] + 180 z - 285 z^(3/2) - 360 z^2 + 606 z^(5/2) + 732 z^3 + 1155 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^4 (1 + Sqrt[z])^5)) ((40 + 60 Sqrt[z] - 180 z - 285 z^(3/2) + 360 z^2 + 606 z^(5/2) - 732 z^3 + 1155 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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-1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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[FractionBox["19", "4"], ",", FractionBox["19", "4"], ",", FractionBox["9", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["20", "-", RowBox[List["105", " ", "z"]], "+", RowBox[List["246", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1887", " ", 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"]], ")"]], "5"]]]], "-", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["20", "-", RowBox[List["105", " ", "z"]], "+", RowBox[List["246", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1887", " ", 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"]], ")"]], "5"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "40"]], "+", RowBox[List["60", " ", SqrtBox["z"]]], "+", RowBox[List["180", " ", "z"]], "-", RowBox[List["285", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["360", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["606", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["732", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1155", " ", 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"]]], ")"]], "5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "4"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["40", "+", RowBox[List["60", " ", SqrtBox["z"]]], "-", RowBox[List["180", " ", "z"]], "-", RowBox[List["285", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["360", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["606", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["732", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1155", " ", 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"]]], ")"]], "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "5"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], "2"]]], RowBox[List["38115", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["7", "/", "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]