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http://functions.wolfram.com/03.04.06.0014.02
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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]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", 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"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> K </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> s </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mfrac> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> s </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mfrac> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BesselK </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <abs /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <apply> <abs /> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <abs /> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <abs /> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <ci> s </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <apply> <abs /> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <abs /> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <abs /> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
