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http://functions.wolfram.com/04.02.27.0003.01
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Ceiling[z] == Round[z + (1 + I)/2] - CharacteristicFunction[(Re[z] + 1)/2,
Integers] - I CharacteristicFunction[(Im[z] + 1)/2, Integers]
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Cell[BoxData[RowBox[List[RowBox[List["Ceiling", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["Round", "[", RowBox[List["z", "+", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "]"]], "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "+", "1"]]]], "2"], ",", "Integers"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "+", "1"]]]], "2"], ",", "Integers"]], "]"]]]]]]]]]]
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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> <mi> z </mi> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ⌉ </mo> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mo> ⌈ </mo> <mi> z </mi> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ⌉ </mo> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Ceiling", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Round", "[", RowBox[List["z", "+", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "]"]], "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "+", "1"]], ")"]]]], ",", "Integers"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "+", "1"]], ")"]]]], ",", "Integers"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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