Modified Bessel function of the second kind: Integral transforms (formula 03.04.22.0005)

BesselK

$Mathematica Notation$

Bessel-Type Functions

BesselK[nu,z]

Integral transforms

Hankel transforms

http://functions.wolfram.com/03.04.22.0005.01

Input Form

HankelTransform[BesselK[\[Nu], t], {t, \[Mu]}, z] == (1/(Sqrt[2] Gamma[1 + \[Mu]])) (z^(1/2 + \[Mu]) Gamma[(1/4) (3 + 2 \[Mu] - 2 \[Nu])] Gamma[(1/4) (3 + 2 \[Mu] + 2 \[Nu])] Hypergeometric2F1[(1/4) (3 + 2 \[Mu] - 2 \[Nu]), (1/4) (3 + 2 \[Mu] + 2 \[Nu]), 1 + \[Mu], -z^2]) /; Re[\[Mu] - \[Nu]] > -(3/2) && Re[\[Mu] + \[Nu]] > -(3/2) && z > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HankelTransform", "[", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", RowBox[List["{", RowBox[List["t", ",", "\[Mu]"]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["2"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "-", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]], "\[And]", RowBox[List["z", ">", "0"]]]]]]]]

MathML Form

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Rule Form

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

2001-10-29