Integer part of the quotient: Series representations (formula 04.07.06.0002)

Quotient

$Mathematica Notation$

Integer Functions

Quotient[m,n]

Series representations

Other series representations

http://functions.wolfram.com/04.07.06.0002.01

Input Form

Quotient[m, n] == m/n - 1/2 + (1/(2 n)) Sum[Sin[(2 Pi k m)/n] Cot[(Pi k)/n], {k, 1, n - 1}] /; Element[m, Integers] && (Element[n - 1, Integers] && n - 1 > 0) && !Element[m/n, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Quotient", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List[FractionBox["m", "n"], "-", FractionBox["1", "2"], "+", RowBox[List[FractionBox["1", RowBox[List["2", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "k", " ", "m"]], RowBox[List["n", " "]]], "]"]], RowBox[List["Cot", "[", FractionBox[RowBox[List["\[Pi]", " ", "k"]], RowBox[List["n", " "]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["n", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["n", "-", "1"]], ">", "0"]]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[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> <mrow> <mi> quotient </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <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> <mi> quotient </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <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["Quotient", "[", RowBox[List["m_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["m", "n"], "-", FractionBox["1", "2"], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "k", " ", "m"]], "n"], "]"]], " ", RowBox[List["Cot", "[", FractionBox[RowBox[List["\[Pi]", " ", "k"]], "n"], "]"]]]]]], RowBox[List["2", " ", "n"]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["n", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "1"]], ">", "0"]]]], ")"]], "&&", RowBox[List["!", RowBox[List[FractionBox["m", "n"], "\[Element]", "Integers"]]]]]]]]]]]]

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

2001-10-29