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Round




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/04.03.27.0004.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Round", "[", "z", "]"]], "\[Equal]", RowBox[List["z", "-", RowBox[List["FractionalPart", "[", RowBox[List["z", "+", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "]"]], "-", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]]]], ")"]]]], ",", "Integers"]], "]"]]]], "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], ")"]]]], ",", "Integers"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Im", "[", "z", "]"]]]], ",", "Integers"]], "]"]], "+", RowBox[List["UnitStep", "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]]]], "+", RowBox[List["Sign", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Re", "[", "z", "]"]]]], ",", "Integers"]], "]"]], "+", RowBox[List["UnitStep", "[", RowBox[List[FractionBox["1", "2"], "+", 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> <mo> &#8970; </mo> <mi> z </mi> <mo> &#8969; </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </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> <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> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#8970;z&#8969; </ms> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> &#8520; </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> frac </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> &#8520; </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#967; </ms> <ms> &#8484; </ms> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> 4 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> &#8520; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#967; </ms> <ms> &#8484; </ms> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> Im </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> 4 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> sgn </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#967; </ms> <ms> &#8484; </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> &#952; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> &#8520; </ms> <apply> <ci> RowBox </ci> <list> <ms> sgn </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#967; </ms> <ms> &#8484; </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Im </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> &#952; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Im </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2001-10-29