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BesselK






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselK[nu,z] > Representations through more general functions > Through Meijer G > Classical cases involving 0F1





http://functions.wolfram.com/03.04.26.0087.01









  


  










Input Form





BesselK[\[Nu], 2 Sqrt[z]] Hypergeometric0F1[-1 + \[Nu], z] == (Gamma[-1 + \[Nu]]/(4 Sqrt[2])) ((2 Sqrt[2] (-1 + \[Nu]))/z^(\[Nu]/2) + 2^\[Nu] Pi^(3/2) Csc[Pi (-(5/4) + \[Nu])] MeijerG[{{1 - \[Nu]/2, 3/2 - \[Nu]/2}, {5/4 - \[Nu]/2, -(1/4) + \[Nu]/2}}, {{2 - \[Nu]/2, \[Nu]/2}, {2 - (3 \[Nu])/2, 5/4 - \[Nu]/2, -(\[Nu]/2), -(1/4) + \[Nu]/2}}, 4 z])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]], RowBox[List["4", " ", SqrtBox["2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]]]], "+", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["5", "4"]]], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["2", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]]]], ")"]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", RowBox[List["2", " ", SqrtBox["z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]_"]], ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]]], "+", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["5", "4"]]], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["2", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", SqrtBox["2"]]]]]]]]










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





2007-05-02