PHY4604 R. D. Field Department of Physics Chapter2_16.doc University of Florida The Dirac Delta Function Dirac Delta Function: The Dirac delta function is not really a function (mathematically it is called a distribution ). It corresponds to an infinitely high, infinitesimally narrow spike at the point x = a. ∞ = −0 ) ( a x δ a x a x = ≠ with 1 ) ( = − ∫ +∞ ∞ − dx a x Properties: The delta function has the following properties ) ( ) ( ) ( ) ( a x a f a x x f − = − and ) ( ) ( ) ( a f dx a x x f = − ∫ +∞ ∞ − , where f(x) is an ordinary function. One can show that ) ( | | 1 ) ( x c cx = and  ) ( ) ( | | 2 1 ) ( 2 2 c x c x c c x + + − = − , where c is a real constant. Integral Representations: The following is an integral representation of a delta function ( i.e.
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