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Cos




Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Complex characteristics > Signum value





http://functions.wolfram.com/01.07.19.0015.01









  


  








Input Form





Sign[Cos[z]] == Sqrt[2] ((Cos[Re[z]] Cosh[Im[z]] - I Sin[Re[z]] Sinh[Im[z]])/ Sqrt[Cos[2 Re[z]] + Cosh[2 Im[z]]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["Cos", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[SqrtBox["2"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["Im", "[", "z", "]"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["Im", "[", "z", "]"]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "+", RowBox[List["Cosh", "[", 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> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </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> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </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> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <mi> cos </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> cosh </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> </msqrt> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <cos /> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <cosh /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["Cos", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["Im", "[", "z", "]"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["Im", "[", "z", "]"]], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]]]]]]]]


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





2007-05-02