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http://functions.wolfram.com/03.17.06.0012.01
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KelvinBei[\[Nu], z] == Exp[2 \[Nu] Pi I Floor[Arg[z - x]/(2 Pi)]]
Sum[((2^(-1 - 3 (k/2)) (I - 1)^k)/k!)
(Sum[Binomial[k, 2 j] ((-I) (1 - I^k) KelvinBer[4 j - k + \[Nu], x] +
(1 + I^k) KelvinBei[4 j - k + \[Nu], x]), {j, 0, Floor[k/2]}] +
Sum[Binomial[k, 2 j + 1] (I (1 - I^k) KelvinBer[2 + 4 j - k + \[Nu],
x] - (1 + I^k) KelvinBei[2 + 4 j - k + \[Nu], x]),
{j, 0, Floor[(k - 1)/2]}]) (z - x)^k, {k, 0, Infinity}] /;
Element[x, Reals] && x < 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Exp", "[", RowBox[List["2", "\[Nu]", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["3", RowBox[List["k", "/", "2"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", "-", "1"]], ")"]], "k"]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["k", "/", "2"]], "]"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["2", "j"]]]], "]"]], RowBox[List["(", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", "x"]], "]"]]]]]], ")"]], ")"]]]]]], "+", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "/", "2"]], "]"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]], "]"]], RowBox[List["(", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", "x"]], "]"]]]], "-", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", "x"]], "]"]]]]]], ")"]], ")"]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]
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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> bei </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msup> <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> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mi> k </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "j"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> ⅈ </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> bei </mi> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> ⅈ </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> ber </mi> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> ⅈ </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> ber </mi> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> ⅈ </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> bei </mi> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> KelvinBei </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> <pi /> <imaginaryi /> <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> </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> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> k </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> k </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <imaginaryi /> <ci> k </ci> </apply> </apply> <apply> <ci> KelvinBei </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> ν </ci> </apply> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <imaginaryi /> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> KelvinBer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> ν </ci> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> k </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <imaginaryi /> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> KelvinBer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <imaginaryi /> <ci> k </ci> </apply> </apply> <apply> <ci> KelvinBei </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
