Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9un)

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=21/4

http://functions.wolfram.com/07.23.03.a9un.01

Input Form

Hypergeometric2F1[-(19/4), 21/4, 1/2, z] == (1/(2197845 Pi^(3/2))) ((-8 Sqrt[z] (11764845 - 181040640 z + 734877696 z^2 - 1086324736 z^3 + 528482304 z^4) EllipticE[(1/2) (1 - Sqrt[z])] + 8 Sqrt[z] (11764845 - 181040640 z + 734877696 z^2 - 1086324736 z^3 + 528482304 z^4) EllipticE[(1/2) (1 + Sqrt[z])] + (2197845 + 47059380 Sqrt[z] - 86362560 z - 724162560 z^(3/2) + 509859840 z^2 + 2939510784 z^(5/2) - 937689088 z^3 - 4345298944 z^(7/2) + 528482304 z^4 + 2113929216 z^(9/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (2197845 - 47059380 Sqrt[z] - 86362560 z + 724162560 z^(3/2) + 509859840 z^2 - 2939510784 z^(5/2) - 937689088 z^3 + 4345298944 z^(7/2) + 528482304 z^4 - 2113929216 z^(9/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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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"]]], ",", FractionBox["21", "4"], ",", FractionBox["1", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["11764845", "-", RowBox[List["181040640", " ", "z"]], "+", RowBox[List["734877696", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1086324736", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List["8", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["11764845", "-", RowBox[List["181040640", " ", "z"]], "+", RowBox[List["734877696", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1086324736", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2197845", "+", RowBox[List["47059380", " ", SqrtBox["z"]]], "-", RowBox[List["86362560", " ", "z"]], "-", RowBox[List["724162560", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["509859840", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["2939510784", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["937689088", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["4345298944", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["2113929216", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2197845", "-", RowBox[List["47059380", " ", SqrtBox["z"]]], "-", RowBox[List["86362560", " ", "z"]], "+", RowBox[List["724162560", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["509859840", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2939510784", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["937689088", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["4345298944", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["2113929216", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]]], ")"]], " ", 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["2197845", " ", 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]