Dirac Delta FunctionDate: 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/ |
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