Coth[z + Pi I] == Coth[z]

Cell[BoxData[RowBox[List[RowBox[List["Coth", "[", RowBox[List["z", "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]], "]"]], "\[Equal]", RowBox[List["Coth", "[", "z", "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <coth /> <apply> <plus /> <ci> z </ci> <apply> <times /> <pi /> <imaginaryi /> </apply> </apply> </apply> <apply> <coth /> <ci> z </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Coth", "[", RowBox[List["z_", "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Coth", "[", "z", "]"]]]]]]

2007-05-02