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

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=-27/8, b>=a

For fixed z and a=-27/8, b=-1/8

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

Input Form

Hypergeometric2F1[-(27/8), -(1/8), 2, z] == (1/(197505 Pi z)) (8 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (627 + 34319 z + 1221 z^2 - 387 z^3 + 60 z^4) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (8151 + 1118 z - 369 z^2 + 60 z^3) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 20 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-1254 - 661 z + 162 z^2 - 45 z^3 + 6 z^4) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (627 + 34319 z + 1221 z^2 - 387 z^3 + 60 z^4) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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<times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </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["27", "8"]]], ",", RowBox[List["-", FractionBox["1", "8"]]], ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["8", " ", SuperscriptBox["2", RowBox[List["3", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List["627", "+", RowBox[List["34319", " ", "z"]], "+", RowBox[List["1221", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["387", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["60", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "+", RowBox[List["3", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List["8151", "+", RowBox[List["1118", " ", "z"]], "-", RowBox[List["369", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["60", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "+", RowBox[List["20", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1254"]], "-", RowBox[List["661", " ", "z"]], "+", RowBox[List["162", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["45", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["6", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["627", "+", RowBox[List["34319", " ", "z"]], "+", RowBox[List["1221", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["387", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["60", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["197505", " ", "\[Pi]", " ", "z"]]]]]]]

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]