Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aj6v)

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.aj6v.01

Input Form

Hypergeometric2F1[3/4, 19/4, 1/2, z] == (1/(4620 Pi^(3/2))) (((2 (-1155 - 1770 z + 1405 z^2 - 656 z^3 + 128 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^5 + (2 (-1155 - 1770 z + 1405 z^2 - 656 z^3 + 128 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^5 + (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^4)) ((1155 + 1080 Sqrt[z] + 690 z - 1000 z^(3/2) - 405 z^2 + 560 z^(5/2) + 96 z^3 - 128 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^4 (1 + Sqrt[z])^5)) ((1155 - 1080 Sqrt[z] + 690 z + 1000 z^(3/2) - 405 z^2 - 560 z^(5/2) + 96 z^3 + 128 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> <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["3", "4"], ",", FractionBox["19", "4"], ",", FractionBox["1", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1155"]], "-", RowBox[List["1770", " ", "z"]], "+", RowBox[List["1405", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["656", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["128", " ", 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"]], ")"]], "5"]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1155"]], "-", RowBox[List["1770", " ", "z"]], "+", RowBox[List["1405", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["656", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["128", " ", 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"]], ")"]], "5"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1155", "+", RowBox[List["1080", " ", SqrtBox["z"]]], "+", RowBox[List["690", " ", "z"]], "-", RowBox[List["1000", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["405", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["560", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["96", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["128", " ", 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["1155", "-", RowBox[List["1080", " ", SqrtBox["z"]]], "+", RowBox[List["690", " ", "z"]], "+", RowBox[List["1000", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["405", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["560", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["96", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["128", " ", 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["4620", " ", 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]