Product[Sech[z/2^k], {k, 1, Infinity}] == z Csch[z]

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["Sech", "[", FractionBox["z", SuperscriptBox["2", "k"]], "]"]]]], "\[Equal]", RowBox[List["z", " ", RowBox[List["Csch", "[", "z", "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mi> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sech /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], RowBox[List["Sech", "[", FractionBox["z_", SuperscriptBox["2", "k_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["z", " ", RowBox[List["Csch", "[", "z", "]"]]]]]]]]

2007-05-02