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   http://functions.wolfram.com/03.16.27.0003.01
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    KelvinKer[z] == Piecewise[
  {{(-I) Pi BesselI[0, (-1)^(1/4) z] + (1/2) BesselK[0, (-1)^(1/4) z] - 
     (1/4) Pi (3 I BesselJ[0, (-1)^(1/4) z] + BesselY[0, (-1)^(1/4) z]), 
    Inequality[(3 Pi)/4, Less, Arg[z], LessEqual, Pi]}}, 
  (1/2) BesselK[0, (-1)^(1/4) z] - (1/4) Pi ((-I) BesselJ[0, (-1)^(1/4) z] + 
     BesselY[0, (-1)^(1/4) z])] 
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   Cell[BoxData[RowBox[List[RowBox[List["KelvinKer", "[", "z", "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["BesselI", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", RowBox[List["BesselJ", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], ")"]]]]]], ",", RowBox[List[FractionBox[RowBox[List["3", " ", "\[Pi]"]], "4"], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]], "}"]], "}"]], ",", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], ")"]]]]]]]], "]"]]]]]] 
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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>  <mi> ker </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo>  </mo>  <mrow>  <mo>  </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mi> ⅈ </mi>  </mrow>  <mo> ⁢ </mo>  <mi> π </mi>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> I </mi>  <mn> 0 </mn>  </msub>  <mo> ( </mo>  <mrow>  <mroot>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 4 </mn>  </mroot>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> K </mi>  <mn> 0 </mn>  </msub>  <mo> ( </mo>  <mrow>  <mroot>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 4 </mn>  </mroot>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  <mo> ⁢ </mo>  <mi> π </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> J </mi>  <mn> 0 </mn>  </msub>  <mo> ( </mo>  <mrow>  <mroot>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 4 </mn>  </mroot>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <msub>  <mi> Y </mi>  <mn> 0 </mn>  </msub>  <mo> ( </mo>  <mrow>  <mroot>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 4 </mn>  </mroot>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mtd>  <mtd>  <mrow>  <mfrac>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mn> 4 </mn>  </mfrac>  <mo> < </mo>  <mrow>  <mi> arg </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ≤ </mo>  <mi> π </mi>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> K </mi>  <mn> 0 </mn>  </msub>  <mo> ( </mo>  <mrow>  <mroot>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 4 </mn>  </mroot>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  <mo> ⁢ </mo>  <mi> π </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <msub>  <mi> Y </mi>  <mn> 0 </mn>  </msub>  <mo> ( </mo>  <mrow>  <mroot>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 4 </mn>  </mroot>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> - </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> J </mi>  <mn> 0 </mn>  </msub>  <mo> ( </mo>  <mrow>  <mroot>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 4 </mn>  </mroot>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mtd>  <mtd>  <semantics>  <mi> True </mi>  <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation>  </semantics>  </mtd>  </mtr>  </mtable>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <ci> KelvinKer </ci>  <ci> z </ci>  </apply>  <piecewise>  <piece>  <apply>  <plus />  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <imaginaryi />  </apply>  <pi />  <apply>  <ci> BesselI </ci>  <cn type='integer'> 0 </cn>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='rational'> 1 <sep /> 2 </cn>  <apply>  <ci> BesselK </ci>  <cn type='integer'> 0 </cn>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='rational'> 1 <sep /> 4 </cn>  <pi />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 3 </cn>  <imaginaryi />  <apply>  <ci> BesselJ </ci>  <cn type='integer'> 0 </cn>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <ci> BesselY </ci>  <cn type='integer'> 0 </cn>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <ci> Inequality </ci>  <apply>  <times />  <cn type='integer'> 3 </cn>  <pi />  <apply>  <power />  <cn type='integer'> 4 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <lt />  <apply>  <arg />  <ci> z </ci>  </apply>  <leq />  <pi />  </apply>  </piece>  <otherwise>  <apply>  <plus />  <apply>  <times />  <cn type='rational'> 1 <sep /> 2 </cn>  <apply>  <ci> BesselK </ci>  <cn type='integer'> 0 </cn>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='rational'> 1 <sep /> 4 </cn>  <pi />  <apply>  <plus />  <apply>  <ci> BesselY </ci>  <cn type='integer'> 0 </cn>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <ci> z </ci>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <ci> BesselJ </ci>  <cn type='integer'> 0 </cn>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </otherwise>  </piecewise>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["BesselI", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", RowBox[List["BesselJ", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], ")"]]]]]], RowBox[List[FractionBox[RowBox[List["3", " ", "\[Pi]"]], "4"], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]], List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List["0", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], ")"]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]]  |  
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   Date Added to functions.wolfram.com (modification date)
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