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FractionalPart






Mathematica Notation

Traditional Notation









Integer Functions > FractionalPart[z] > Transformations > Multiple arguments





http://functions.wolfram.com/04.05.16.0012.01









  


  










Input Form





FractionalPart[n z] == n FractionalPart[z] - Sum[k (UnitStep[Mod[z, 1] - k/n] (1 - UnitStep[Mod[z, 1] - (k + 1)/n])), {k, 0, n - 1}] + n - 1 + Sign[UnitStep[z] + CharacteristicFunction[n z, Integers]] - n Sign[UnitStep[z] + CharacteristicFunction[z, Integers]] /; Element[n, Integers] && n >= 0 && Element[z, Reals]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FractionalPart", "[", RowBox[List["n", " ", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["n", " ", RowBox[List["FractionalPart", "[", "z", "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["z", ",", "1"]], "]"]], "-", FractionBox["k", "n"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["z", ",", "1"]], "]"]], "-", FractionBox[RowBox[List["k", "+", "1"]], "n"]]], "]"]]]], ")"]]]], ")"]]]]]], "+", "n", "-", "1", "+", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", "z", "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["n", " ", "z"]], ",", "Integers"]], "]"]]]], "]"]], "-", RowBox[List["n", " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", "z", "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List["z", ",", "Integers"]], "]"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["z", ",", "Reals"]], "]"]]]]]]]]










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> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mi> n </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> <mtext> </mtext> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <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> <mtext> </mtext> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`z </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> - </mo> <mfrac> <mi> k </mi> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`z </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> - </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> FractionalPart </ci> <apply> <times /> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> FractionalPart </ci> <ci> z </ci> </apply> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <ci> z </ci> </apply> <apply> <ci> UnitStep </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <times /> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <ci> UnitStep </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <ci> k </ci> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <rem /> <ci> $CellContext`z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <rem /> <ci> $CellContext`z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> z </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FractionalPart", "[", RowBox[List["n_", " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["n", " ", RowBox[List["FractionalPart", "[", "z", "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["z", ",", "1"]], "]"]], "-", FractionBox["k", "n"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["z", ",", "1"]], "]"]], "-", FractionBox[RowBox[List["k", "+", "1"]], "n"]]], "]"]]]], ")"]]]], ")"]]]]]], "+", "n", "-", "1", "+", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", "z", "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["n", " ", "z"]], ",", "Integers"]], "]"]]]], "]"]], "-", RowBox[List["n", " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", "z", "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List["z", ",", "Integers"]], "]"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["z", "\[Element]", "Reals"]]]]]]]]]]










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





2001-10-29





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