Sign[z^a] == Abs[z]^a /; Element[I a, Reals]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sign", "[", SuperscriptBox["z", "a"], "]"]], "\[Equal]", SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], "a"]]], " ", "/;", " ", RowBox[List["Element", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], ",", "Reals"]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </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> <mi> a </mi> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <abs /> <ci> z </ci> </apply> <ci> a </ci> </apply> </apply> <apply> <in /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", SuperscriptBox["z_", "a_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], "a"], "/;", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "\[Element]", "Reals"]]]]]]]]

2001-10-29