Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.cdl0)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a>0

For fixed z and a=41/8, b>=a

For fixed z and a=41/8, b=43/8

http://functions.wolfram.com/07.23.03.cdl0.01

Input Form

Hypergeometric2F1[41/8, 43/8, 3, z] == (256 2^(1/4) (16 Sqrt[1 - z] (-323 + 5814 z + 359136 z^2 + 493098 z^3 + 74115 z^4) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 8 Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (-323 + 5814 z + 359136 z^2 + 493098 z^3 + 74115 z^4) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (-1 + z) (-2584 + 45543 z + 922491 z^2 + 833205 z^3 + 65025 z^4) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 8 Sqrt[1 - z] (-323 + 5814 z + 359136 z^2 + 493098 z^3 + 74115 z^4) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (251818875 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] (-1 + z)^8 z^2)

Standard Form

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MathML Form

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type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["41", "8"], ",", FractionBox["43", "8"], ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["256", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["16", " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "323"]], "+", RowBox[List["5814", " ", "z"]], "+", RowBox[List["359136", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["493098", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["74115", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "+", RowBox[List["8", " ", SqrtBox[RowBox[List["2", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "323"]], "+", RowBox[List["5814", " ", "z"]], "+", RowBox[List["359136", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["493098", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["74115", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2584"]], "+", RowBox[List["45543", " ", "z"]], "+", RowBox[List["922491", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["833205", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["65025", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List["8", " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "323"]], "+", RowBox[List["5814", " ", "z"]], "+", RowBox[List["359136", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["493098", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["74115", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["251818875", " ", "\[Pi]", " ", SqrtBox[RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", SuperscriptBox["z", "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]