KelvinBei[z] == Sum[(((-1 + I)^k 2^(-1 - (3 k)/2))/k!) (Sum[Binomial[k, 2 j] ((1 + I^k) KelvinBei[4 j - k, Subscript[z, 0]] - I (1 - I^k) KelvinBer[4 j - k, Subscript[z, 0]]), {j, 0, Floor[k/2]}] + Sum[Binomial[k, 1 + 2 j] ((-1 - I^k) KelvinBei[2 + 4 j - k, Subscript[z, 0]] + I (1 - I^k) KelvinBer[2 + 4 j - k, Subscript[z, 0]]), {j, 0, Floor[(k - 1)/2]}]) (z - Subscript[z, 0])^k, {k, 0, Infinity}]

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</mo> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> &#8520; </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> bei </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> &#8520; </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> ber </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; 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2007-05-02