Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











FractionalPart






Mathematica Notation

Traditional Notation









Integer Functions > FractionalPart[z] > Representations through equivalent functions > With related functions > With Round > For complex arguments





http://functions.wolfram.com/04.05.27.0003.01









  


  










Input Form





FractionalPart[z] == z - Round[z - (1 + I)/2] - 1 - I - CharacteristicFunction[(Re[z] + 1)/2, Integers] - I CharacteristicFunction[(Im[z] + 1)/2, Integers] + Sign[UnitStep[Re[z]] + CharacteristicFunction[Re[z], Integers]] + I Sign[UnitStep[Im[z]] + CharacteristicFunction[Im[z], Integers]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["FractionalPart", "[", "z", "]"]], "\[Equal]", RowBox[List["z", "-", RowBox[List["Round", "[", RowBox[List["z", "-", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "]"]], "-", "1", "-", "\[ImaginaryI]", "-", 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"]], "]"]]]], "+", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]]]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8969; </mo> </mrow> <mo> - </mo> <mn> 1 </mn> <mo> - </mo> <mi> &#8520; </mi> <mo> - </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </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> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8969; </mo> </mrow> <mo> - </mo> <mn> 1 </mn> <mo> - </mo> <mi> &#8520; </mi> <mo> - </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </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> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <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> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FractionalPart", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["z", "-", RowBox[List["Round", "[", RowBox[List["z", "-", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "]"]], "-", "1", "-", "\[ImaginaryI]", "-", 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"]], "]"]]]], "+", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]]]], "]"]]]]]]]]]]










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





2001-10-29