Imaginary part: Transformations (formula 12.04.16.0017)

$Mathematica Notation$

Complex Components

Im[z]

Transformations

Power of arguments

http://functions.wolfram.com/12.04.16.0017.01

Input Form

Im[z^a] == Abs[z]^a Sin[a Arg[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["Arg", "[", "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> ⁡ </mo> <mo> ( </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", 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> <arg /> <ci> z </ci> </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["Arg", "[", "z", "]"]]]], "]"]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]]]

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

2001-10-29