Integrate[t^(\[Alpha] - 1) Quotient[t, n], {t, 0, a}] == (1/\[Alpha]) (a^\[Alpha] Quotient[a, n] + n^\[Alpha] (-Zeta[-\[Alpha]] + Zeta[-\[Alpha], 1 + Quotient[a, n]]))

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

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> a </mi> </msubsup> <mrow> <msup> <mi> t </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> &#945; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mi> &#945; </mi> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> n </mi> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, &quot;\[Alpha]&quot;]], Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;quotient&quot;, &quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;,&quot;, &quot;n&quot;]], &quot;)&quot;]], &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`x, $CellContext`y], Zeta[$CellContext`x, $CellContext`y]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;\[Alpha]&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <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> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> quotient </ci> <ci> t </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> quotient </ci> <ci> a </ci> <ci> n </ci> </apply> <apply> <power /> <ci> a </ci> <ci> &#945; </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> n </ci> <ci> &#945; </ci> </apply> <apply> <plus /> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <plus /> <apply> <ci> quotient </ci> <ci> a </ci> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

2001-10-29