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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/04.03.21.0007.01









  


  








Input Form





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








Standard Form





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








MathML Form







<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> &#8747; </mo> <mi> a </mi> <mi> &#8734; </mi> </msubsup> <mrow> <msup> <mi> t </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mi> t </mi> <mo> &#8969; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> &#945; </mi> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mi> &#945; </mi> </msup> </mrow> <mo> + </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </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;\[LeftFloor]&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], &quot;\[RightFloor]&quot;]], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`x, $CellContext`y], Zeta[$CellContext`x, $CellContext`y]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> &#8747; </ms> <ms> a </ms> <ms> &#8734; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> t </ms> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> &#8970;t&#8969; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> t </ms> </list> </apply> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> &#945; </ms> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <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> <apply> <ci> SuperscriptBox </ci> <ms> a </ms> <ms> &#945; </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </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> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> &#945; </ms> <ms> ) </ms> </list> </apply> <ms> &lt; </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </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]", "a_", "\[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[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List["Floor", "[", RowBox[List["a", "+", FractionBox["1", "2"]]], "]"]]]], "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["Floor", "[", RowBox[List["a", "+", FractionBox["1", "2"]]], "]"]]]]]], "]"]]]], "\[Alpha]"]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", "1"]]]]]]]]]]


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





2001-10-29