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 Ceiling > For complex arguments





http://functions.wolfram.com/04.05.27.0021.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["FractionalPart", "[", "z", "]"]], "\[Equal]", RowBox[List["z", "-", RowBox[List["Ceiling", "[", "z", "]"]], "+", RowBox[List["Sign", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", "Integers"]], "]"]], "+", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]], "+", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "z", "]"]], "]"]]]], "]"]], "-", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ",", "Integers"]], "]"]]]], "]"]], "+", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["Re", "[", "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> &#8968; </mo> <mi> z </mi> <mo> &#8969; </mo> </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> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <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> <mo> ( </mo> <mrow> <mn> 1 </mn> <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> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <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> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <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> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> FractionalPart </ci> <ci> z </ci> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <ci> z </ci> </apply> </apply> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> &#967; </ci> <ci> &#8484; </ci> </apply> <apply> <real /> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2007-05-02