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Quotient




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/04.07.27.0003.01









  


  








Input Form





Quotient[m, n] == Round[m/n - (1 + I)/2] + I CharacteristicFunction[(1/2) (1 + Im[m/n]), Integers] + CharacteristicFunction[(1/2) (1 + Re[m/n]), Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Quotient", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Round", "[", RowBox[List[FractionBox["m", "n"], "-", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], RowBox[List[" ", "2"]]]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[" ", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Im", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "Integers"]], "]"]]]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[" ", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Re", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "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> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8969; </mo> </mrow> <mo> + </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8969; </mo> </mrow> <mo> + </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Quotient", "[", RowBox[List["m_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Round", "[", RowBox[List[FractionBox["m", "n"], "-", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Im", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "Integers"]], "]"]]]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Re", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "Integers"]], "]"]]]]]]]]


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





2001-10-29