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http://functions.wolfram.com/04.06.21.0012.01
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Integrate[Mod[m, t], {t, 0, a}] ==
(a m + a Mod[m, a] - m^2 PolyGamma[1, (a + m - Mod[m, a])/a])/2
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Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a"], " ", RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["a", " ", "m"]], "+", RowBox[List["a", " ", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "-", RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox[RowBox[List["a", "+", "m", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "a"]]], "]"]]]]]], "2"]]]]]
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<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> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`t </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> <mrow> <mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <semantics> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`a </ci> </apply> </annotation-xml> </semantics> </mrow> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <semantics> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`a </ci> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> </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`m </ci> <ci> $CellContext`t </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`a </ci> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> m </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`a </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a_"], RowBox[List[RowBox[List["Mod", "[", RowBox[List["m_", ",", "t_"]], "]"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "m"]], "+", RowBox[List["a", " ", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "-", RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox[RowBox[List["a", "+", "m", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "a"]]], "]"]]]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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