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Floor




Mathematica Notation

Traditional Notation









Integer Functions > Floor[z] > Transformations > Products, sums, and powers of the direct function > Sums of the direct function





http://functions.wolfram.com/04.01.16.0019.01









  


  








Input Form





Sum[Floor[(k m + x)/n], {k, 0, n - 1}] == Sum[Floor[(k n + x)/m], {k, 0, m - 1}] /; Element[x, Reals] && Element[n, Integers] && n > 0 && Element[m, Integers] && m > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["k", " ", "m"]], "+", "x"]], "n"], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["k", " ", "n"]], "+", "x"]], "m"], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]








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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> x </mi> </mrow> <mi> n </mi> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> x </mi> </mrow> <mi> m </mi> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> m </ci> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> n </ci> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], RowBox[List["n_", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["k_", " ", "m_"]], "+", "x_"]], "n_"], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["k", " ", "n"]], "+", "x"]], "m"], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]


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





2001-10-29