DiracDelta[x] == Limit[(1/(Pi x)) Sin[x/\[CurlyEpsilon]], \[CurlyEpsilon] -> 0]

Cell[BoxData[RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["\[Pi]", " ", "x"]]], RowBox[List["Sin", "[", " ", FractionBox["x", RowBox[List[" ", "\[CurlyEpsilon]"]]], "]"]]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", "0"]]]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'><mrow><mrow><semantics><mi>δ</mi><annotation-xml encoding='MathML-Content'><ci>DiracDelta</ci></annotation-xml></semantics><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩵</mo><mrow><munder><mi>lim</mi><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></munder><mrow><mo>⁢</mo><mtext> </mtext><mfrac><mn>1</mn><mrow><mi>π</mi><mo>⁢</mo><mi>x</mi></mrow></mfrac><mo>⁢</mo><mrow><mi>sin</mi><mo>⁡</mo><mo>(</mo><mfrac><mi>x</mi><mi>ϵ</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow></math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DiracDelta", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["Sin", "[", FractionBox["x", "\[CurlyEpsilon]"], "]"]], RowBox[List["\[Pi]", " ", "x"]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", "0"]]]], "]"]]]]]]

2001-10-29