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Mod




Mathematica Notation

Traditional Notation









Integer Functions > Mod[m,n] > Representations through equivalent functions > With related functions > With Ceiling > For complex arguments





http://functions.wolfram.com/04.06.27.0018.01









  


  








Input Form





Mod[m, n] == m - n Ceiling[m/n] - n UnitStep[CharacteristicFunction[Re[m/n], Integers] - 1] + I n UnitStep[-CharacteristicFunction[Im[m/n], Integers]] + n








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List["m", "-", RowBox[List["n", " ", RowBox[List["Ceiling", "[", FractionBox["m", "n"], "]"]]]], "-", RowBox[List["n", " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", FractionBox["m", "n"], "]"]], ",", "Integers"]], "]"]], "-", "1"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "n", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", FractionBox["m", "n"], "]"]], ",", "Integers"]], "]"]]]], "]"]]]], "+", "n"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`m </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> &#8968; </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> &#8969; </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mtext> </mtext> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mtext> </mtext> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <rem /> <ci> FE`Conversion`Private`m </ci> <ci> $CellContext`n </ci> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ceiling /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <real /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> n </ci> <imaginaryi /> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <imaginary /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <ci> n </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Mod", "[", RowBox[List["m_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["m", "-", RowBox[List["n", " ", RowBox[List["Ceiling", "[", FractionBox["m", "n"], "]"]]]], "-", RowBox[List["n", " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", FractionBox["m", "n"], "]"]], ",", "Integers"]], "]"]], "-", "1"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "n", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", FractionBox["m", "n"], "]"]], ",", "Integers"]], "]"]]]], "]"]]]], "+", "n"]]]]]]


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





2007-05-02