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Cos




Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Transformations > Argument involving complex components





http://functions.wolfram.com/01.07.16.0082.01









  


  








Input Form





Cos[a Arg[z]] == ((1/2) ((Re[z] - (I Abs[Im[z]] Abs[Re[z]])/Re[z])^a + (Re[z] + (I Abs[Im[z]] Abs[Re[z]])/Re[z])^a))/Abs[z]^a








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Cos", "[", RowBox[List["a", " ", RowBox[List["Arg", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], RowBox[List["-", "a"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], RowBox[List["Re", "[", "z", "]"]]]]], ")"]], "a"], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], RowBox[List["Re", "[", "z", "]"]]]]], ")"]], "a"]]], ")"]]]]]]]]








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> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <cos /> <apply> <times /> <ci> a </ci> <apply> <arg /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <abs /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> a </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cos", "[", RowBox[List["a_", " ", RowBox[List["Arg", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], RowBox[List["-", "a"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], RowBox[List["Re", "[", "z", "]"]]]]], ")"]], "a"], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], RowBox[List["Re", "[", "z", "]"]]]]], ")"]], "a"]]], ")"]]]]]]]]


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





2001-10-29