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BesselK






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselK[nu,z] > Series representations > Residue representations





http://functions.wolfram.com/03.04.06.0014.02









  


  










Input Form





BesselK[n, z] == (1/2) Sum[Residue[(Gamma[s + Abs[n]/2]/(z/2)^(2 s)) Gamma[s - Abs[n]/2], {s, Abs[n]/2 - j}], {j, 0, Abs[n] - 1}] + (1/2) Sum[Residue[(Gamma[s + Abs[n]/2] Gamma[s - Abs[n]/2])/(z/2)^(2 s), {s, -(Abs[n]/2) - j}], {j, 0, Infinity}] /; Element[n, Integers]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "-", "1"]]], RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["Abs", "[", "n", "]"]], "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox[RowBox[List["Abs", "[", "n", "]"]], "2"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[FractionBox[RowBox[List["Abs", "[", "n", "]"]], "2"], "-", "j"]]]], "}"]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["Abs", "[", "n", "]"]], "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "-", FractionBox[RowBox[List["Abs", "[", "n", "]"]], "2"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Abs", "[", "n", "]"]], "2"]]], "-", "j"]]]], "}"]]]], "]"]]]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]










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





2001-10-29





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