Csc[z] == -2 I E^(I z) Sum[E^(2 I z k), {k, 0, Infinity}] /; Im[z] > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Csc", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", "k"]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ">", "0"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <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> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </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> <csc /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <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> <imaginaryi /> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Csc", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", "k"]]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ">", "0"]]]]]]]]

2001-10-29