|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/04.01.27.0003.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Floor[z] == Ceiling[z] + UnitStep[CharacteristicFunction[Re[z], Integers] -
1] - I UnitStep[-CharacteristicFunction[Im[z], Integers]] - 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["Floor", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["Ceiling", "[", "z", "]"]], "+", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]], "-", "1"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]]]], "]"]]]], "-", "1"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ⌊ </mo> <mi> z </mi> <mo> ⌋ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ⌈ </mo> <mi> z </mi> <mo> ⌉ </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mtext> </mtext> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mtext> </mtext> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <floor /> <ci> z </ci> </apply> <apply> <plus /> <apply> <ceiling /> <ci> z </ci> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> χ </ci> <ci> ℤ </ci> </apply> <apply> <real /> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <apply> <ci> Subscript </ci> <ci> χ </ci> <ci> ℤ </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Floor", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Ceiling", "[", "z", "]"]], "+", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]], "-", "1"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]]]], "]"]]]], "-", "1"]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|