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variants of this functions
KelvinKer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKer[z] > Series representations > Generalized power series > Expansions on branch cuts





http://functions.wolfram.com/03.16.06.0006.01









  


  










Input Form





KelvinKer[z] \[Proportional] -2 I Pi Floor[Arg[-x + z]/(2 Pi)] KelvinBer[x] + KelvinKer[x] - (1/Sqrt[2]) (-2 I Pi Floor[Arg[-x + z]/(2 Pi)] (KelvinBei[1, x] + KelvinBer[1, x]) + KelvinKei[1, x] + KelvinKer[1, x]) (x - z) + (1/4) (2 I Pi Floor[Arg[-x + z]/(2 Pi)] (KelvinBei[x] - KelvinBei[2, x]) - KelvinKei[x] + KelvinKei[2, x]) (x - z)^2 + \[Ellipsis] /; (z -> x) && Element[x, Reals] && x < 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", "z", "]"]], "\[Proportional]", "\n", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBer", "[", "x", "]"]]]], "+", RowBox[List["KelvinKer", "[", "x", "]"]], "-", RowBox[List[FractionBox["1", SqrtBox["2"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["1", ",", "x"]], "]"]], "+", RowBox[List["KelvinBer", "[", RowBox[List["1", ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List["KelvinKei", "[", RowBox[List["1", ",", "x"]], "]"]], "+", RowBox[List["KelvinKer", "[", RowBox[List["1", ",", "x"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["x", "-", "z"]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["KelvinBei", "[", "x", "]"]], "-", RowBox[List["KelvinBei", "[", RowBox[List["2", ",", "x"]], "]"]]]], ")"]]]], "-", RowBox[List["KelvinKei", "[", "x", "]"]], "+", RowBox[List["KelvinKei", "[", RowBox[List["2", ",", "x"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["x", "-", "z"]], ")"]], "2"]]], " ", "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]










MathML Form







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</mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </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> <ci> KelvinKer </ci> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> KelvinBer </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> KelvinBei </ci> <cn type='integer'> 1 </cn> <ci> x </ci> </apply> <apply> <ci> KelvinBer </ci> <cn type='integer'> 1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <ci> KelvinKei </ci> <cn type='integer'> 1 </cn> <ci> x </ci> </apply> <apply> <ci> KelvinKer </ci> <cn type='integer'> 1 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> KelvinBei </ci> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KelvinBei </ci> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KelvinKei </ci> <ci> x </ci> </apply> </apply> <apply> <ci> KelvinKei </ci> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <ci> x </ci> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </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["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBer", "[", "x", "]"]]]], "+", RowBox[List["KelvinKer", "[", "x", "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["1", ",", "x"]], "]"]], "+", RowBox[List["KelvinBer", "[", RowBox[List["1", ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List["KelvinKei", "[", RowBox[List["1", ",", "x"]], "]"]], "+", RowBox[List["KelvinKer", "[", RowBox[List["1", ",", "x"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["x", "-", "z"]], ")"]]]], SqrtBox["2"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["KelvinBei", "[", "x", "]"]], "-", RowBox[List["KelvinBei", "[", RowBox[List["2", ",", "x"]], "]"]]]], ")"]]]], "-", RowBox[List["KelvinKei", "[", "x", "]"]], "+", RowBox[List["KelvinKei", "[", RowBox[List["2", ",", "x"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["x", "-", "z"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]










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





2007-05-02





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