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Coth






Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Series representations > Dirichlet series





http://functions.wolfram.com/01.22.06.0008.01









  


  










Input Form





Coth[z] == 1 + 2 Sum[E^(-2 z (k + 1)), {k, 0, Infinity}] /; Re[z] > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Coth", "[", "z", "]"]], "\[Equal]", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "z", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]










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> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <coth /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> z </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Coth", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "z", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]










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





2001-10-29