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Round






Mathematica Notation

Traditional Notation









Integer Functions > Round[z] > Integration > Definite integration > For the direct function itself





http://functions.wolfram.com/04.03.21.0006.01









  


  










Input Form





Integrate[t^(\[Alpha] - 1) Round[t], {t, 0, a}] == (1/\[Alpha]) (a^\[Alpha] Floor[a + 1/2] - (1/2 + Floor[a - 1/2])^\[Alpha] + (1 - 2^(-\[Alpha])) Zeta[-\[Alpha]] + Zeta[-\[Alpha], 1/2 + Floor[a - 1/2]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Round", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "\[Alpha]"], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List["Floor", "[", RowBox[List["a", "+", FractionBox["1", "2"]]], "]"]]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Floor", "[", RowBox[List["a", "-", FractionBox["1", "2"]]], "]"]]]], ")"]], "\[Alpha]"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "\[Alpha]"]], "]"]]]], "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Floor", "[", RowBox[List["a", "-", FractionBox["1", "2"]]], "]"]]]]]], "]"]]]], ")"]]]]]]]]










MathML Form







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</ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> &#945; </ms> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#8970; </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> &#8971; </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> $CellContext`x </ci> </bvar> <bvar> <ci> $CellContext`y </ci> </bvar> <apply> <ci> Zeta </ci> <ci> $CellContext`x </ci> <ci> $CellContext`y </ci> </apply> </lambda> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





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["Round", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List["Floor", "[", RowBox[List["a", "+", FractionBox["1", "2"]]], "]"]]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Floor", "[", RowBox[List["a", "-", FractionBox["1", "2"]]], "]"]]]], ")"]], "\[Alpha]"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "\[Alpha]"]], "]"]]]], "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Floor", "[", RowBox[List["a", "-", FractionBox["1", "2"]]], "]"]]]]]], "]"]]]], "\[Alpha]"]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29





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