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

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=-15/4, b>=a

For fixed z and a=-15/4, b=1/4

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

Input Form

Hypergeometric2F1[-(15/4), 1/4, -(9/2), z] == (1/(3360 Pi^(3/2))) (((2 Sqrt[z] (-1680 + 700 z + 291 z^2 + 176 z^3 + 128 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z) - (2 Sqrt[z] (-1680 + 700 z + 291 z^2 + 176 z^3 + 128 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z) + (1/(-1 + Sqrt[z])) ((-3360 + 5040 Sqrt[z] - 1120 z + 420 z^(3/2) - 510 z^2 + 219 z^(5/2) - 272 z^3 + 96 z^(7/2) - 128 z^4) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/(1 + Sqrt[z])) ((3360 + 5040 Sqrt[z] + 1120 z + 420 z^(3/2) + 510 z^2 + 219 z^(5/2) + 272 z^3 + 96 z^(7/2) + 128 z^4) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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</semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["15", "4"]]], ",", FractionBox["1", "4"], ",", RowBox[List["-", FractionBox["9", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1680"]], "+", RowBox[List["700", " ", "z"]], "+", RowBox[List["291", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["176", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["128", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], "-", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1680"]], "+", RowBox[List["700", " ", "z"]], "+", RowBox[List["291", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["176", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["128", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3360"]], "+", RowBox[List["5040", " ", SqrtBox["z"]]], "-", RowBox[List["1120", " ", "z"]], "+", RowBox[List["420", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["510", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["219", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["272", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["96", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["128", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3360", "+", RowBox[List["5040", " ", SqrtBox["z"]]], "+", RowBox[List["1120", " ", "z"]], "+", RowBox[List["420", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["510", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["219", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["272", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["96", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["128", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List["1", "+", SqrtBox["z"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["3360", " ", 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]