Absolute value function: Representations through equivalent functions (formula 12.01.27.0008)

Abs

$Mathematica Notation$

Complex Components

Abs[z]

Representations through equivalent functions

With related functions

With Re

http://functions.wolfram.com/12.01.27.0008.01

Input Form

Abs[z] == Sqrt[-z^2 + 2 z Re[z]]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Equal]", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["Re", "[", "z", "]"]]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⩵ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["Re", "[", "z", "]"]]]]]]]]]]]

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

2007-05-02