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variants of this functions
KelvinKer




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.20.06.0024.01









  


  








Input Form





KelvinKer[2, z] \[Proportional] 1/2 - ((Pi z^2)/32) (1 - z^4/384 + z^8/2211840 + \[Ellipsis]) - (z^4/16) ((1/12) (-(17/6) + 2 EulerGamma + 2 Log[z/2]) - (1/23040) (-(247/60) + 2 EulerGamma + 2 Log[z/2]) z^4 + (1/309657600) (-(512/105) + 2 EulerGamma + 2 Log[z/2]) z^8 + \[Ellipsis]) /; (z -> 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["2", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", "2"], "-", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], "32"], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], "384"], "+", FractionBox[SuperscriptBox["z", "8"], "2211840"], "+", "\[Ellipsis]"]], ")"]]]], " ", "-", RowBox[List[FractionBox[SuperscriptBox["z", "4"], "16"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["17", "6"]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "23040"], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["247", "60"]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]]]], ")"]], SuperscriptBox["z", "4"]]], "+", RowBox[List[FractionBox["1", "309657600"], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["512", "105"]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]]]], ")"]], SuperscriptBox["z", "8"]]], "+", "\[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> <msub> <mi> ker </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 32 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mn> 384 </mn> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 8 </mn> </msup> <mn> 2211840 </mn> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 17 </mn> <mn> 6 </mn> </mfrac> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 247 </mn> <mn> 60 </mn> </mfrac> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mn> 23040 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 512 </mn> <mn> 105 </mn> </mfrac> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> <mtext> </mtext> </mrow> </mrow> <mn> 309657600 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <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> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 32 </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'> 384 </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'> 2211840 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; 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</ci> </apply> </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", "[", RowBox[List["2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", RowBox[List[FractionBox["1", "32"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], "384"], "+", FractionBox[SuperscriptBox["z", "8"], "2211840"], "+", "\[Ellipsis]"]], ")"]]]], "-", RowBox[List[FractionBox["1", "16"], " ", SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["17", "6"]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["247", "60"]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]]]], ")"]], " ", SuperscriptBox["z", "4"]]], "23040"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["512", "105"]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]]]], ")"]], " ", SuperscriptBox["z", "8"]]], "309657600"], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]


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





2007-05-02