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Coth






Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Complex characteristics > Argument





http://functions.wolfram.com/01.22.19.0005.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Arg", "[", RowBox[List["Coth", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sign", "[", RowBox[List[FractionBox[RowBox[List["Sign", "[", RowBox[List["Sin", "[", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "]"]], "]"]], RowBox[List["Sign", "[", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "]"]], "-", RowBox[List["Cosh", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]]]], "]"]]], "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Sign", "[", RowBox[List["Sinh", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "]"]]]], RowBox[List["Sign", "[", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "-", RowBox[List["Cos", "[", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Csch", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]]]]]], ")"]]]]]]]]










MathML Form







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










Rule Form





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










Date Added to functions.wolfram.com (modification date)





2001-10-29





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