Sign[E^z] == E^(I Im[z])

Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", SuperscriptBox["\[ExponentialE]", "z"], "]"]], "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Im", "[", "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> &#8519; </mi> <mi> z </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", SuperscriptBox["\[ExponentialE]", "z_"], "]"]], "]"]], "\[RuleDelayed]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Im", "[", "z", "]"]]]]]]]]]

2001-10-29