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variants of this functions
UnitStep




Mathematica Notation

Traditional Notation









Generalized Functions > UnitStep[x] > Limit representations





http://functions.wolfram.com/14.01.09.0001.01









  


  








Input Form





UnitStep[x] == Limit[1/(1 + Exp[-(x/\[CurlyEpsilon])]), \[CurlyEpsilon] -> Plus[0]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["UnitStep", "[", "x", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[FractionBox["1", RowBox[List["1", "+", RowBox[List["Exp", "[", RowBox[List["-", FractionBox["x", "\[CurlyEpsilon]"]]], "]"]]]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#949; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mi> x </mi> <mi> &#949; </mi> </mfrac> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> UnitStep </ci> <ci> x </ci> </apply> <apply> <limit /> <bvar> <ci> &#949; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#949; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> &#949; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["UnitStep", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[FractionBox["1", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["x", "\[CurlyEpsilon]"]]]]]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]]]]]]


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





2001-10-29