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http://functions.wolfram.com/03.04.06.0026.01
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BesselK[\[Nu], z] == (2^(-2 \[Nu] - 1) Pi^(3/2) Csc[Pi \[Nu]]
Sum[(2^k/(x^k k!)) ((16^\[Nu] Gamma[1 - \[Nu]]
HypergeometricPFQRegularized[{(1 - \[Nu])/2, 1 - \[Nu]/2},
{(1 - \[Nu] - k)/2, (2 - \[Nu] - k)/2, 1 - \[Nu]}, x^2/4])/
E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) -
x^(2 \[Nu]) E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)])
Gamma[\[Nu] + 1] HypergeometricPFQRegularized[{(\[Nu] + 1)/2,
(\[Nu] + 2)/2}, {(1 + \[Nu] - k)/2, (2 + \[Nu] - k)/2, \[Nu] + 1},
x^2/4]) (z - x)^k, {k, 0, Infinity}])/x^\[Nu] /;
!Element[\[Nu], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["x", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "k"], " ", SuperscriptBox["x", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["16", "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]", "-", "k"]], "2"], ",", FractionBox[RowBox[List["2", "-", "\[Nu]", "-", "k"]], "2"], ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]], ",", FractionBox[SuperscriptBox["x", "2"], "4"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["x", RowBox[List["2", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ",", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]", "-", "k"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]", "-", "k"]], "2"], ",", RowBox[List["\[Nu]", "+", "1"]]]], "}"]], ",", FractionBox[RowBox[List[" ", SuperscriptBox["x", "2"]]], "4"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "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> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mo> - </mo> <mi> k </mi> 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<apply> <times /> <cn type='integer'> -2 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <power /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 16 </cn> <ci> ν </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 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</semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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