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http://functions.wolfram.com/04.01.27.0031.01
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Floor[z] == z - FractionalPart[z] - I /;
(Element[I z, Reals] && I z > 0 && !Element[I z, Integers]) ||
(Re[z] > 0 && Im[z] < 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Floor", "[", "z", "]"]], "\[Equal]", RowBox[List["z", "-", RowBox[List["FractionalPart", "[", "z", "]"]], "-", "\[ImaginaryI]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], ">", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Integers"]], "]"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "<", "0"]]]], ")"]]]]]]]]
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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> <mrow> <mo> ⌊ </mo> <mi> z </mi> <mo> ⌋ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> frac </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <floor /> <ci> z </ci> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> FractionalPart </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <reals /> </apply> <apply> <gt /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <notin /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <integers /> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Floor", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", "-", RowBox[List["FractionalPart", "[", "z", "]"]], "-", "\[ImaginaryI]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], ">", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Integers"]]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "<", "0"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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