Congruence function: General characteristics (formula 04.06.04.0003)

Mod

$Mathematica Notation$

Integer Functions

Mod[m,n]

General characteristics

Sets of discontinuity

http://functions.wolfram.com/04.06.04.0003.01

Input Form

DiscontinuitySets[Mod[m, n], m] == {SequenceList[{Interval[{k n - I Infinity, k n + I Infinity}], -1}, Element[k, Integers]], SequenceList[ {Interval[{I k n - Infinity, I k n + Infinity}], -I}, Element[k, Integers]]}

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["DiscontinuitySets", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "n"]], "]"]], ",", "m"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["k", " ", "n"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], ",", RowBox[List[RowBox[List["k", " ", "n"]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]]], "}"]], "]"]], ",", RowBox[List["-", "1"]]]], "}"]], ",", RowBox[List["k", "\[Element]", "Integers"]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "k", " ", "n"]], "-", "\[Infinity]"]], ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "k", " ", "n"]], "+", "\[Infinity]"]]]], "}"]], "]"]], ",", RowBox[List["-", "\[ImaginaryI]"]]]], "}"]], ",", RowBox[List["k", "\[Element]", "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> <msub> <mi> 𝒟𝒮 </mi> <mi> m </mi> </msub> <mo> ( </mo> <semantics> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> ∞ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> 𝒟𝒮 </mi> <mi> m </mi> </msub> <mo> ( </mo> <semantics> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> ∞ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DiscontinuitySets", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["m_", ",", "n_"]], "]"]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["{", RowBox[List[RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["k", " ", "n"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], ",", RowBox[List[RowBox[List["k", " ", "n"]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]]], "}"]], "]"]], ",", RowBox[List["-", "1"]]]], "}"]], ",", RowBox[List["k", "\[Element]", "Integers"]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "k", " ", "n"]], "-", "\[Infinity]"]], ",", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "k", " ", "n"]], "+", "\[Infinity]"]]]], "}"]], "]"]], ",", RowBox[List["-", "\[ImaginaryI]"]]]], "}"]], ",", RowBox[List["k", "\[Element]", "Integers"]]]], "]"]]]], "}"]]]]]]

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

2001-10-29