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variants of this functions
KelvinKei






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKei[nu,z] > Series representations > Residue representations





http://functions.wolfram.com/03.19.06.0048.01









  


  










Input Form





KelvinKei[\[Nu], z] == (-(1/4)) Sum[Residue[((Gamma[s - \[Nu]/4] Gamma[s + (\[Nu] + 2)/4] Gamma[s + (2 - \[Nu])/4])/((z/4)^(4 s) (Gamma[\[Nu]/2 + s] Gamma[1 - \[Nu]/2 - s]))) Gamma[s + \[Nu]/4], {s, -j - \[Nu]/4}], {j, 0, Infinity}] - (1/4) Sum[Residue[((Gamma[s + \[Nu]/4] Gamma[s + (\[Nu] + 2)/4] Gamma[s + (2 - \[Nu])/4])/((z/4)^(4 s) (Gamma[\[Nu]/2 + s] Gamma[1 - \[Nu]/2 - s]))) Gamma[s - \[Nu]/4], {s, -j + \[Nu]/4}], {j, 0, Infinity}] - (1/4) Sum[Residue[((Gamma[s + \[Nu]/4] Gamma[s - \[Nu]/4] Gamma[s + (2 - \[Nu])/4])/((z/4)^(4 s) (Gamma[\[Nu]/2 + s] Gamma[1 - \[Nu]/2 - s]))) Gamma[s + (\[Nu] + 2)/4], {s, -j - (\[Nu] + 2)/4}], {j, 0, Infinity}] - (1/4) Sum[Residue[((Gamma[s + \[Nu]/4] Gamma[s - \[Nu]/4] Gamma[s + (2 + \[Nu])/4])/((z/4)^(4 s) (Gamma[\[Nu]/2 + s] Gamma[1 - \[Nu]/2 - s]))) Gamma[s + (2 - \[Nu])/4], {s, -j - (2 - \[Nu])/4}], {j, 0, Infinity}] /; !Element[\[Nu], Integers]










Standard Form





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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> kei </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; 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</mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; 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</mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> KelvinKei </ci> <ci> &#957; 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Rule Form





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





2007-05-02





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