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variants of this functions
KelvinKer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKer[z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself





http://functions.wolfram.com/03.16.06.0009.01









  


  










Input Form





KelvinKer[z] \[Proportional] (-Log[z/2]) (1 - z^4/64 + z^8/147456 + \[Ellipsis]) + (-EulerGamma + ((2 EulerGamma - 3)/128) z^4 - ((-25 + 12 EulerGamma)/1769472) z^8 + \[Ellipsis]) + ((Pi z^2)/16) (1 - z^4/576 + z^8/3686400 + \[Ellipsis]) /; (z -> 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], "64"], "+", FractionBox[SuperscriptBox["z", "8"], "147456"], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "EulerGamma"]], "-", "3"]], "128"], " ", SuperscriptBox["z", "4"]]], "-", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List["-", "25"]], "+", RowBox[List["12", " ", "EulerGamma"]]]], ")"]], "1769472"], SuperscriptBox["z", "8"]]], "+", "\[Ellipsis]"]], ")"]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], "16"], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], "576"], "+", FractionBox[SuperscriptBox["z", "8"], "3686400"], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> ker </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> log </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mn> 64 </mn> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 8 </mn> </msup> <mn> 147456 </mn> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <semantics> <mrow> <mi> &#8509; </mi> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8509; </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mn> 128 </mn> </mfrac> <mo> &#8290; </mo> <mtext> </mtext> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[DoubledGamma]&quot;, &quot;+&quot;, RowBox[List[FractionBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]]]], &quot;-&quot;, &quot;3&quot;]], &quot;128&quot;], &quot; &quot;, SuperscriptBox[&quot;z&quot;, &quot;4&quot;]]]]], Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 25 </mn> </mrow> <mn> 1769472 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mn> 576 </mn> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 8 </mn> </msup> <mn> 3686400 </mn> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> KelvinKer </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> log </ci> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 64 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <cn type='integer'> 147456 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <eulergamma /> </apply> <cn type='integer'> -25 </cn> </apply> <apply> <power /> <cn type='integer'> 1769472 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 576 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <cn type='integer'> 3686400 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], "64"], "+", FractionBox[SuperscriptBox["z", "8"], "147456"], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List[FractionBox["1", "128"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "EulerGamma"]], "-", "3"]], ")"]], " ", SuperscriptBox["z", "4"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "25"]], "+", RowBox[List["12", " ", "EulerGamma"]]]], ")"]], " ", SuperscriptBox["z", "8"]]], "1769472"], "+", "\[Ellipsis]"]], ")"]], "+", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], "576"], "+", FractionBox[SuperscriptBox["z", "8"], "3686400"], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]










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





2007-05-02





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