Abs[z^a] == E^(Re[a] Log[Abs[z]] - Im[a] Arg[z])

Cell[BoxData[RowBox[List[RowBox[List["Abs", "[", SuperscriptBox["z", "a"], "]"]], "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]]]], "-", RowBox[List[RowBox[List["Im", "[", "a", "]"]], " ", RowBox[List["Arg", "[", "z", "]"]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msup> <mi> z </mi> <mi> a </mi> </msup> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <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> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <exp /> <apply> <plus /> <apply> <times /> <apply> <real /> <ci> a </ci> </apply> <apply> <ln /> <apply> <abs /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <imaginary /> <ci> a </ci> </apply> <apply> <arg /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", SuperscriptBox["z_", "a_"], "]"]], "]"]], "\[RuleDelayed]", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], " ", RowBox[List["Log", "[", RowBox[List["Abs", "[", "z", "]"]], "]"]]]], "-", RowBox[List[RowBox[List["Im", "[", "a", "]"]], " ", RowBox[List["Arg", "[", "z", "]"]]]]]]]]]]]

2001-10-29