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Power




Mathematica Notation

Traditional Notation









Elementary Functions > Power[z,a] > Complex characteristics > Imaginary part





http://functions.wolfram.com/01.02.19.0012.01









  


  








Input Form





Im[z^a] == Abs[z]^a Sin[a ArcTan[Re[z], Im[z]]] /; Element[a, Reals]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", SuperscriptBox["z", "a"], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], "a"], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List["a", ",", "Reals"]], "]"]]]]]]








MathML Form







<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> z </mi> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <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> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <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> <imaginary /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <abs /> <ci> z </ci> </apply> <ci> a </ci> </apply> <apply> <sin /> <apply> <times /> <ci> a </ci> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", SuperscriptBox["z_", "a_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], "a"], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29