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 FractionalPart

 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

 frac ( n z ) frac ( z ) n - n sgn ( χ ( z ) + θ UnitStep ( z ) ) + sgn ( χ ( n z ) + θ UnitStep ( z ) ) - k = 0 n - 1 k θ UnitStep ( z mod 1 \$CellContext`z 1 - k n ) ( 1 - θ UnitStep ( z mod 1 \$CellContext`z 1 - k + 1 n ) ) + n - 1 /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] z TagBox["\[DoubleStruckCapitalR]", Function[Reals]] Condition FractionalPart n z FractionalPart z n -1 n Sign Subscript χ z UnitStep z Sign Subscript χ n z UnitStep z -1 k 0 n -1 k UnitStep \$CellContext`z 1 -1 k n -1 1 -1 UnitStep \$CellContext`z 1 -1 k 1 n -1 n -1 n z [/itex]

 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