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Ceiling




Mathematica Notation

Traditional Notation









Integer Functions > Ceiling[z] > Transformations > Transformations and argument simplifications > Argument involving related functions





http://functions.wolfram.com/04.02.16.0021.01









  


  








Input Form





Ceiling[FractionalPart[z]] == (1 - CharacteristicFunction[Re[-z], Integers]) UnitStep[Re[z]] + (1 - CharacteristicFunction[Im[-z], Integers]) UnitStep[Im[z]] I








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Ceiling", "[", RowBox[List["FractionalPart", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", RowBox[List["-", "z"]], "]"]], ",", "Integers"]], "]"]]]], ")"]], RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", RowBox[List["-", "z"]], "]"]], ",", "Integers"]], "]"]]]], ")"]], RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "\[ImaginaryI]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8968; </mo> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8969; </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mtext> </mtext> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mtext> </mtext> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ceiling /> <apply> <ci> FractionalPart </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <imaginary /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <real /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Ceiling", "[", RowBox[List["FractionalPart", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", RowBox[List["-", "z"]], "]"]], ",", "Integers"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", RowBox[List["-", "z"]], "]"]], ",", "Integers"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", "\[ImaginaryI]"]]]]]]]]


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





2007-05-02