Nearest integer function: Representations through equivalent functions (formula 04.03.27.0007)

Round

$Mathematica Notation$

Integer Functions

Round[z]

Representations through equivalent functions

With elementary functions

http://functions.wolfram.com/04.03.27.0007.01

Input Form

Round[z] == z - ArcTan[Tan[Pi z]]/Pi /; Element[z, Reals] && !Element[z + 1/2, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Round", "[", "z", "]"]], "\[Equal]", RowBox[List["z", "-", FractionBox[RowBox[List["ArcTan", "[", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]], "\[Pi]"]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["z", ",", "Reals"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["z", "+", FractionBox["1", "2"]]], ",", "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> <mrow> <mo> ⌊ </mo> <mi> z </mi> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> tan </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> π </mi> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mi> z </mi> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> tan </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> π </mi> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Round", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List["ArcTan", "[", RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]], "\[Pi]"]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["z", "+", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]]]]]

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

2001-10-29