Im[x^a] == x^Re[a] Sin[Im[a] Log[x]] /; Element[x, Reals] && x > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", SuperscriptBox["x", "a"], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["x", RowBox[List["Re", "[", "a", "]"]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["Im", "[", "a", "]"]], " ", RowBox[List["Log", "[", "x", "]"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["x", ",", "Reals"]], "]"]], "\[And]", RowBox[List["x", ">", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> x </mi> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> x </mi> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <imaginary /> <apply> <power /> <ci> x </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> x </ci> <apply> <real /> <ci> a </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <imaginary /> <ci> a </ci> </apply> <apply> <ln /> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", SuperscriptBox["x_", "a_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["x", RowBox[List["Re", "[", "a", "]"]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["Im", "[", "a", "]"]], " ", RowBox[List["Log", "[", "x", "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "0"]]]]]]]]]]

2001-10-29