Dirac Delta Function

Date: 05/15/2000 at 19:54:08
From: Betty
Subject: Dirac's delta function

How is one to interpret Dirac's delta function? What is it, and what 
does it mean? It is used in proofs in a text on stochastic 
differential equations I'm reading, and I can't find a description of 
what it represents in any of the texts available to me. (I live in a 
very rural area, and have to rely on my own library.)

Date: 05/15/2000 at 23:05:42
From: Doctor Douglas
Subject: Re: Dirac's delta function

Hi Betty,

Thanks for writing to Ask Dr. Math.

The Dirac delta function is a special kind of object - it is not a 
function in the usual sense, although we sometimes think of it that 
way. Basically, it can be thought of as the "function" d(x) such that:

     d(x) = 0         for x <> 0
          = Infinity  for x =  0

and the "size" of the infinity is such that the integral of d(x) with 
respect to x from a to b is zero if a and b are of the same sign, and 
the integral is exactly unity if a < 0 < b. More precisely, it can be 
thought of as the limit of a series of functions that get more and 
more peaked near zero, and whose area is always unity.

It is always to be implicitly used inside an integral, even though we 
carry out operations such as:

     f(x) = d(x) + d(x-2)     (when integrated, this has area 2)

You can read more about the Dirac delta function from Eric Weisstein's
MathWorld site here:

  Delta Function
  http://mathworld.wolfram.com/DeltaFunction.html   

If you have more questions, please feel free to write back!

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/