Cosh[z + 2 m I Pi] == Cosh[z] /; Element[m, Integers]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mtext> </mtext> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <cosh /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> m </ci> </apply> </apply> </apply> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", RowBox[List["z_", "+", RowBox[List["2", " ", "m_", " ", "\[ImaginaryI]", " ", "\[Pi]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], "/;", RowBox[List["m", "\[Element]", "Integers"]]]]]]]]

2001-10-29