E^(I Arg[z]) == Cos[Arg[z]] + I Sin[Arg[z]]

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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> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#10869; </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <arg /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <cos /> <apply> <arg /> <ci> z </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <arg /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Arg", "[", "z_", "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Cos", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]]]]]]]]]]

2001-10-29