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variants of this functions
KelvinBer






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.18.06.0056.01









  


  










Input Form





KelvinBer[\[Nu], z] == Pi Sum[Residue[(Gamma[s + (2 + \[Nu])/4]/((z/4)^(4 s) (Gamma[s + \[Nu] + 1/2] Gamma[1/2 - s - \[Nu]] Gamma[-s + (2 + \[Nu])/4] Gamma[1 - s + \[Nu]/4]))) Gamma[s + \[Nu]/4], {s, -j - \[Nu]/4}], {j, 0, Infinity}] + Pi Sum[Residue[(Gamma[s + \[Nu]/4]/((z/4)^(4 s) (Gamma[s + \[Nu] + 1/2] Gamma[1/2 - s - \[Nu]] Gamma[-s + (2 + \[Nu])/4] Gamma[1 - s + \[Nu]/4]))) Gamma[s + (2 + \[Nu])/4], {s, -j - (\[Nu] + 2)/4}], {j, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["\[Pi]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["\[Nu]", "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List["\[Pi]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]]]], "}"]]]], "]"]]]]]]]]]]]]










MathML Form







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</mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </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> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <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> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> KelvinBer </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -4 </cn> <ci> s </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -4 </cn> <ci> s </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["\[Nu]", "4"]]]]], "}"]]]], "]"]]]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", FractionBox["\[Nu]", "4"]]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "j"]], "-", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]]]], "}"]]]], "]"]]]]]]]]]]]]










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





2007-05-02





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