Im[E^z] == E^Re[z] Sin[Im[z]]

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

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

2007-05-02