Arg[Cosh[z]] == ArcTan[Tan[Im[z]] Tanh[Re[z]]] + (1/2) Pi (1 - Sign[Cos[Im[z]] Cosh[Re[z]]]) Sign[Sign[Sin[Im[z]] Sinh[Re[z]]] + 1/2]

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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> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arg /> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <arctan /> <apply> <times /> <apply> <tan /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <tanh /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <ci> Sign </ci> <apply> <times /> <apply> <sin /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <times /> <apply> <cos /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <cosh /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Arg", "[", RowBox[List["Cosh", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Tan", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List[RowBox[List["Cos", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["Sign", "[", RowBox[List[RowBox[List["Sin", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "]"]], "+", FractionBox["1", "2"]]], "]"]]]]]]]]]]

2001-10-29