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http://functions.wolfram.com/04.03.21.0008.01
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Integrate[t^(\[Alpha] - 1) Round[t], {t, 1, Infinity}] ==
-((1 + Zeta[-\[Alpha], 3/2])/\[Alpha]) /; Re[\[Alpha]] < -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Round", "[", "t", "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["1", "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", FractionBox["3", "2"]]], "]"]]]], "\[Alpha]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", "1"]]]]]]]]
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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> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> <mrow> <msup> <mi> t </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ⌊ </mo> <mi> t </mi> </mrow> <mo> ⌉ </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> α </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "\[Alpha]"]], Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[List[$CellContext`a, $CellContext`b], Zeta[$CellContext`a, $CellContext`b]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </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> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> <mrow> <msup> <mi> t </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ⌊ </mo> <mi> t </mi> </mrow> <mo> ⌉ </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> α </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "\[Alpha]"]], Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[List[$CellContext`a, $CellContext`b], Zeta[$CellContext`a, $CellContext`b]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </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> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Round", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["1", "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", FractionBox["3", "2"]]], "]"]]]], "\[Alpha]"]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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