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

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.andh.01

Input Form

Hypergeometric2F1[19/4, 19/4, 5/2, z] == (1/(88935 Pi^(3/2) z^(3/2))) ((-((2 Sqrt[z] (45 + 20985 z + 37767 z^2 + 6739 z^3) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^7) - (2 Sqrt[z] (45 + 20985 z + 37767 z^2 + 6739 z^3) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^7 + (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^6)) ((-90 + 135 Sqrt[z] + 2430 z + 18555 z^(3/2) + 10770 z^2 + 26997 z^(5/2) + 3274 z^3 + 3465 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^6 (1 + Sqrt[z])^7)) ((90 + 135 Sqrt[z] - 2430 z + 18555 z^(3/2) - 10770 z^2 + 26997 z^(5/2) - 3274 z^3 + 3465 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'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3465 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3274 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 26997 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 10770 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 18555 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2430 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 135 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn 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</apply> <cn type='integer'> 90 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </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["5", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["45", "+", RowBox[List["20985", " ", "z"]], "+", RowBox[List["37767", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["6739", " ", 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"]], ")"]], "7"]]]], "-", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["45", "+", RowBox[List["20985", " ", "z"]], "+", RowBox[List["37767", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["6739", " ", 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"]], ")"]], "7"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "90"]], "+", RowBox[List["135", " ", SqrtBox["z"]]], "+", RowBox[List["2430", " ", "z"]], "+", RowBox[List["18555", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["10770", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["26997", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["3274", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["3465", " ", 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"]]], ")"]], "7"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "6"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["90", "+", RowBox[List["135", " ", SqrtBox["z"]]], "-", RowBox[List["2430", " ", "z"]], "+", RowBox[List["18555", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["10770", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["26997", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["3274", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["3465", " ", 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"]]], ")"]], "6"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "7"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], "2"]]], RowBox[List["88935", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", 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]