Congruence function: Integration (formula 04.06.21.0007)

Mod

$Mathematica Notation$

For the direct function with respect to m

http://functions.wolfram.com/04.06.21.0007.01

Input Form

Integrate[Mod[t, n], {t, 0, a}] == (1/2) (a n - n Mod[a, n] + Mod[a, n]^2)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["a", " "]]], RowBox[List[RowBox[List["Mod", "[", RowBox[List["t", ",", "n"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "n"]], "-", RowBox[List["n", " ", RowBox[List["Mod", "[", RowBox[List["a", ",", "n"]], "]"]]]], "+", SuperscriptBox[RowBox[List["Mod", "[", RowBox[List["a", ",", "n"]], "]"]], "2"]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> a </mi> </msubsup> <mrow> <semantics> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`t </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <semantics> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`a </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <semantics> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`a </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> a </ci> </uplimit> <apply> <rem /> <ci> $CellContext`t </ci> <ci> $CellContext`n </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <rem /> <ci> $CellContext`a </ci> <ci> $CellContext`n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <rem /> <ci> $CellContext`a </ci> <ci> $CellContext`n </ci> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a_"], RowBox[List[RowBox[List["Mod", "[", RowBox[List["t_", ",", "n_"]], "]"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "n"]], "-", RowBox[List["n", " ", RowBox[List["Mod", "[", RowBox[List["a", ",", "n"]], "]"]]]], "+", SuperscriptBox[RowBox[List["Mod", "[", RowBox[List["a", ",", "n"]], "]"]], "2"]]], ")"]]]]]]]]

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

2001-10-29