Modified Bessel function of the second kind: Integral representations (formula 03.04.07.0012)

BesselK

$Mathematica Notation$

Bessel-Type Functions

BesselK[nu,z]

Integral representations

Contour integral representations

http://functions.wolfram.com/03.04.07.0012.01

Input Form

BesselK[\[Nu], z] == ((z^2)^(\[Nu]/2) (1/(4 Pi I)) ContourIntegral[(Gamma[s + \[Nu]/2] Gamma[s - \[Nu]/2])/(z^2/4)^s, \[ScriptCapitalL]["s"]])/z^\[Nu] - (Pi (z^(2 \[Nu]) - (z^2)^\[Nu]) Csc[Pi \[Nu]] (1/(4 Pi I)) ContourIntegral[Gamma[s + \[Nu]/2]/Gamma[1 + \[Nu]/2 - s]/(-(z^2/4))^s, \[ScriptCapitalL]["s"]])/((-z)^(\[Nu]/2) z^((3 \[Nu])/2)) /; !Element[\[Nu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["\[Nu]", "/", "2"]]], FractionBox["1", RowBox[List["4", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubscriptBox["\[Integral]", "\[ScriptCapitalL]"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]], "-", RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "\[Nu]"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], FractionBox["1", RowBox[List["4", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubscriptBox["\[Integral]", "\[ScriptCapitalL]"], RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]

MathML Form

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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["ContourIntegral", "[", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["\[ScriptCapitalL]", "[", "\"s\"", "]"]]]], "]"]]]], RowBox[List["4", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "-", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", FractionBox["\[Nu]", "2"]]]], " ", SuperscriptBox["z", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["ContourIntegral", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]]], ",", RowBox[List["\[ScriptCapitalL]", "[", "\"s\"", "]"]]]], "]"]]]], RowBox[List["4", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29