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 DiracDelta

 http://functions.wolfram.com/14.03.07.0002.01

 Input Form

 DiracDelta[x] == (1/Pi) Integrate[E^(I t x) UnitStep[t], {t, -Infinity, Infinity}] - I/(Pi x)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "\[Pi]"], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "t", " ", "x"]]], RowBox[List["UnitStep", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["\[Pi]", " ", "x"]]]]]]]]]

 MathML Form

 δ DiracDelta ( x ) 1 π - t x θ UnitStep ( t ) t - π x DiracDelta x 1 -1 t -1 t x UnitStep t -1 x -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DiracDelta", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "t", " ", "x"]]], " ", RowBox[List["UnitStep", "[", "t", "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Pi]"], "-", FractionBox["\[ImaginaryI]", RowBox[List["\[Pi]", " ", "x"]]]]]]]]]

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

 2001-10-29