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Ceiling




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/04.02.21.0007.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "a", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Ceiling", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "\[Alpha]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List["Ceiling", "[", "a", "]"]]]], "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List["Ceiling", "[", "a", "]"]]]], "]"]]]], ")"]]]]]], "/;", 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> <mrow> <msup> <mi> t </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8968; </mo> <mi> t </mi> <mo> &#8969; </mo> </mrow> </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> &#8968; </mo> <mi> a </mi> <mo> &#8969; </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> <mo> &#8968; </mo> <mi> a </mi> <mo> &#8969; </mo> </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[&quot;\[LeftCeiling]&quot;, &quot;a&quot;, &quot;\[RightCeiling]&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> 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> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ceiling /> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ceiling /> <ci> a </ci> </apply> <apply> <power /> <ci> a </ci> <ci> &#945; </ci> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <ceiling /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <real /> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </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["Ceiling", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List["Ceiling", "[", "a", "]"]]]], "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List["Ceiling", "[", "a", "]"]]]], "]"]]]], "\[Alpha]"]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", "1"]]]]]]]]]]


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





2001-10-29