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http://functions.wolfram.com/04.06.21.0014.01
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Integrate[t^(\[Alpha] - 1) Mod[m, t], {t, a, Infinity}] ==
(-(1/(\[Alpha] (1 + \[Alpha])))) (a^\[Alpha] \[Alpha] Mod[m, a] +
m (a^\[Alpha] + m^\[Alpha] \[Alpha] Zeta[1 + \[Alpha]] -
m^\[Alpha] \[Alpha] Zeta[1 + \[Alpha], (a + m - Mod[m, a])/a])) /;
Re[\[Alpha]] < 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "a", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Mod", "[", RowBox[List["m", ",", "t"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "\[Alpha]"], "+", RowBox[List[SuperscriptBox["m", "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Zeta", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]]]], "-", RowBox[List[SuperscriptBox["m", "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", FractionBox[RowBox[List["a", "+", "m", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "a"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "0"]]]]]]
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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> <mrow> <msubsup> <mo> ∫ </mo> <mi> a </mi> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </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> 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> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> α </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> <mo> ⁢ </mo> <msup> <mi> a </mi> <mi> α </mi> </msup> </mrow> <mo> + </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mi> α </mi> </msup> <mo> + </mo> <mrow> <msup> <mi> m </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["\[Alpha]", "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> m </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["\[Alpha]", "+", "1"]], Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "m", "-", InterpretationBox[RowBox[List["m", " ", "mod", " ", "a"]], Mod[$CellContext`m, $CellContext`a]]]], "a"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`x, $CellContext`y], Zeta[$CellContext`x, $CellContext`y]]]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <ci> a </ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <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> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`a </ci> </apply> <apply> <power /> <ci> a </ci> <ci> α </ci> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <power /> <ci> a </ci> <ci> α </ci> </apply> <apply> <times /> <apply> <power /> <ci> m </ci> <ci> α </ci> </apply> <ci> α </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> m </ci> <ci> α </ci> </apply> <ci> α </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <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> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "a_", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Mod", "[", RowBox[List["m_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "\[Alpha]"], "+", RowBox[List[SuperscriptBox["m", "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Zeta", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]]]], "-", RowBox[List[SuperscriptBox["m", "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", FractionBox[RowBox[List["a", "+", "m", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "a"]], "]"]]]], "a"]]], "]"]]]]]], ")"]]]]]], RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
