Sign[z^a] == z^a/E^Re[a Log[z]]

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

<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> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <exp /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

2001-10-29