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http://functions.wolfram.com/04.06.21.0005.01
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Integrate[n^(\[Alpha] - 1) Mod[m, n], n] ==
(n^\[Alpha] (m + \[Alpha] Mod[m, n]))/(\[Alpha] (1 + \[Alpha]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["n", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Mod", "[", RowBox[List["m", ",", "n"]], "]"]], RowBox[List["\[DifferentialD]", "n"]]]]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["n", "\[Alpha]"], " ", RowBox[List["(", RowBox[List["m", "+", RowBox[List["\[Alpha]", " ", RowBox[List["Mod", "[", RowBox[List["m", ",", "n"]], "]"]]]]]], ")"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> n </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <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> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> n </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> n </mi> <mi> α </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mrow> <mi> α </mi> <mo> ⁢ </mo> <mrow> <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> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> n </ci> </bvar> <apply> <times /> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> n </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <ci> α </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <ci> α </ci> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`n </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["n_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Mod", "[", RowBox[List["m_", ",", "n_"]], "]"]]]], RowBox[List["\[DifferentialD]", "n_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["n", "\[Alpha]"], " ", RowBox[List["(", RowBox[List["m", "+", RowBox[List["\[Alpha]", " ", RowBox[List["Mod", "[", RowBox[List["m", ",", "n"]], "]"]]]]]], ")"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
