Sum[(z^k Cosh[k x])/k, {k, 1, Infinity}] == Log[1/Sqrt[z^2 - 2 z Cosh[x] + 1]] /; 0 < Im[x] < 2 Pi && Abs[z] < 1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["z", "k"], " ", RowBox[List["Cosh", "[", RowBox[List["k", " ", "x"]], "]"]]]], "k"]]], "\[Equal]", RowBox[List["Log", "[", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["2", " ", "z", " ", RowBox[List["Cosh", "[", "x", "]"]]]], "+", "1"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Im", "[", "x", "]"]], "<", RowBox[List["2", " ", "\[Pi]"]]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <cosh /> <apply> <times /> <ci> k </ci> <ci> x </ci> </apply> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cosh /> <ci> x </ci> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <imaginary /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["z_", "k_"], " ", RowBox[List["Cosh", "[", RowBox[List["k_", " ", "x_"]], "]"]]]], "k_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Log", "[", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["2", " ", "z", " ", RowBox[List["Cosh", "[", "x", "]"]]]], "+", "1"]]]], "]"]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Im", "[", "x", "]"]], "<", RowBox[List["2", " ", "\[Pi]"]]]], "&&", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]]]

2001-10-29