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Round




Mathematica Notation

Traditional Notation









Integer Functions > Round[z] > Representations through equivalent functions > With related functions > With Mod > For complex arguments





http://functions.wolfram.com/04.03.27.0005.01









  


  








Input Form





Round[z] == z - Mod[z + (1 + I)/2, 1] + (1 + I)/2 - CharacteristicFunction[(2 Re[z] - 1)/4, Integers] - I CharacteristicFunction[(2 Im[z] - 1)/4, Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Round", "[", "z", "]"]], "\[Equal]", RowBox[List["z", "-", RowBox[List["Mod", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], ",", "1"]], "]"]], "+", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"], "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "-", "1"]], "4"], ",", "Integers"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "-", "1"]], "4"], ",", "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> <mo> &#8970; </mo> <mi> z </mi> <mo> &#8969; </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> z </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <plus /> <ci> $CellContext`z </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mo> &#8970; </mo> <mi> z </mi> <mo> &#8969; </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> z </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <plus /> <ci> $CellContext`z </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Round", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["z", "-", RowBox[List["Mod", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], ",", "1"]], "]"]], "+", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"], "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "-", "1"]], ")"]]]], ",", "Integers"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "-", "1"]], ")"]]]], ",", "Integers"]], "]"]]]]]]]]]]


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





2001-10-29