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Floor




Mathematica Notation

Traditional Notation









Integer Functions > Floor[z] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations





http://functions.wolfram.com/04.01.16.0002.01









  


  








Input Form





Floor[-z] == -Floor[z] - I Sign[Abs[Im[z]]] - Sign[Abs[Re[z]]] /; !Element[Re[z], Integers] && !Element[Im[z], Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["Floor", "[", "z", "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "]"]]]], "-", RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "]"]]]]]], "/;", " ", RowBox[List[RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]], "]"]]]]]]]]








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> <mo> &#8970; </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> &#8971; </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mi> z </mi> <mo> &#8971; </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203; </mtext> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mtext> &#8203; </mtext> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> Sign </ci> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <apply> <real /> <ci> z </ci> </apply> <integers /> </apply> <apply> <notin /> <apply> <imaginary /> <ci> z </ci> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Floor", "[", RowBox[List["-", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Floor", "[", "z", "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "]"]]]], "-", RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "]"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Element]", "Integers"]]]]]]]]]]]]


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





2001-10-29