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http://functions.wolfram.com/04.02.16.0003.01
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Ceiling[-z] == -Ceiling[z] + I (1 - CharacteristicFunction[Im[z], Integers])
Sign[Abs[Im[z]]] + (1 - CharacteristicFunction[Re[z], Integers])
Sign[Abs[Re[z]]]
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Cell[BoxData[RowBox[List[RowBox[List["Ceiling", "[", RowBox[List["-", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["Ceiling", "[", "z", "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]]]], ")"]], RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]]]], ")"]], RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ⌈ </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mi> z </mi> <mo> ⌉ </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ceiling /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <ci> z </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <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> <ci> Sign </ci> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </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> χ </ci> <ci> ℤ </ci> </apply> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> Sign </ci> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Ceiling", "[", RowBox[List["-", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["Ceiling", "[", "z", "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]]]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]]]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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