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

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=-33/8, b>=a

For fixed z and a=-33/8, b=-27/8

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

Input Form

Hypergeometric2F1[-(33/8), -(27/8), 2, z] == (1/(688304925 Pi z)) (8 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (479655 + 140191369 z + 825291606 z^2 + 881244738 z^3 + 178224915 z^4 + 2251557 z^5) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (28619415 + 195063172 z + 231100650 z^2 + 51387204 z^3 + 750519 z^4) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 20 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (4316895 + 31976132 z + 46670202 z^2 + 17105220 z^3 + 1315743 z^4) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (479655 + 140191369 z + 825291606 z^2 + 881244738 z^3 + 178224915 z^4 + 2251557 z^5) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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</math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["33", "8"]]], ",", RowBox[List["-", FractionBox["27", "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["479655", "+", RowBox[List["140191369", " ", "z"]], "+", RowBox[List["825291606", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["881244738", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["178224915", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["2251557", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", 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["28619415", "+", RowBox[List["195063172", " ", "z"]], "+", RowBox[List["231100650", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["51387204", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["750519", " ", 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["20", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", RowBox[List["(", RowBox[List["4316895", "+", RowBox[List["31976132", " ", "z"]], "+", RowBox[List["46670202", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["17105220", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1315743", " ", 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["479655", "+", RowBox[List["140191369", " ", "z"]], "+", RowBox[List["825291606", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["881244738", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["178224915", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["2251557", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", 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["688304925", " ", "\[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]