|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/14.01.09.0009.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
E^(I \[Lambda] Pi (UnitStep[x] - UnitStep[-x])) ==
Limit[((\[CurlyEpsilon] + I x)/(\[CurlyEpsilon] - I x))^\[Lambda],
\[CurlyEpsilon] -> Plus[0]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Lambda]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", "x", "]"]], "-", RowBox[List["UnitStep", "[", RowBox[List["-", "x"]], "]"]]]], ")"]]]]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[CurlyEpsilon]", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List["\[CurlyEpsilon]", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]]], ")"]], "\[Lambda]"], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> λ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⩵ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ε </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> ε </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> </mrow> <mrow> <mi> ε </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> λ </mi> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> λ </ci> <pi /> <apply> <plus /> <apply> <ci> UnitStep </ci> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <limit /> <bvar> <ci> ε </ci> </bvar> <condition> <apply> <tendsto /> <ci> ε </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> ε </ci> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> ε </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> λ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Lambda]_", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", "x_", "]"]], "-", RowBox[List["UnitStep", "[", RowBox[List["-", "x_"]], "]"]]]], ")"]]]]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[CurlyEpsilon]", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List["\[CurlyEpsilon]", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]]], ")"]], "\[Lambda]"], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|