DIRAC DELTA FUNCTION AS A DISTRIBUTION

MASSACHUSETTS INSTITUTE OF TECHNOLOGYP hysics : Relativistic Quantum Field Theory IProf .Alan GuthMarch 13, 2008 INFORMAL NOTESDIRAC DELTA FUNCTION AS A DISTRIBUTIONWhy the DIRAC DELTA FUNCTION is not a FUNCTION :The DIRAC DELTA FUNCTION (x) is often described by considering a FUNCTION thathas a narrow peak atx= 0, with unit total area under the peak .In the limit as thepeak becomes infinitely narrow, keeping fixed the area under the peak, the functionis sometimes said to approach a DIRAC DELTA FUNCTION .One example of such a limitisg(x) lim 0g (x),( )whereg (x) 1 2 e 12x2/ 2.( )The area underg (x) is 1, for any value of >0, andg (x) approaches 0 as 0for anyxother thanx= , it was pointed out long ago that the DELTA FUNCTION cannot be rigor-ously defined this way.

Starting with a well-behaved (i.e., piecewise continuous and bounded by some power of t) function f (t), we defined the corresponding distribution by T f [ϕ] ≡ ∞ −∞ d tf (t) ϕ (t). (4.17) Then if we write the distribution corresponding to d f/ d t,weget T d f/ d t [ϕ]= d ∞ −∞ d t d f d t ϕ (t) (4.18) Since f (t) is bounded ...

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  Continuous, Delta, Piecewise, Carid, Dirac delta, Piecewise continuous

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