Proof of Dirac delta sifting property.

Lavabug

1. The problem statement, all variables and given/known data
Prove the statement



3. The attempt at a solution
I am clueless as to how I would go about doing this, I know the basic properties. I think it has to do with using epsilon somewhere and taking the limit as epsilon approaches zero, as shown here:
http://www-thphys.physics.ox.ac.uk/p...n/mm/dirac.pdf
but I really have no idea how they're using it. The prof did something similar in class but he used -epsilon to epsilon in the limits of integration to show that the integral of δ(x)f(x) is just f(0).

Pretty mysterious to me, any help is greatly appreciated.

LCKurtz

Two suggestions you might try.

1. If you have the result for f(0) try letting u = t-a in this problem. Or

2. Parrot your prof's proof only using an integral from a-ε to a+ε.

Hurkyl

In your class, how is the dirac delta defined?


The PDF you linked makes a mistake in its definition of the dirac delta, or more accurately a (rather common) omission -- the limit isn't a limit of functions as you learned in calculus class. It's a different sort of limit, whose relevant property is that if [itex]\varphi[/itex] is a test function, then
[tex]\int_{-\infty}^{+\infty} \left( \lim_{\epsilon \to 0}^{\wedge} f_\epsilon(x) \right) \varphi(x) \, dx = \lim_{\epsilon \to 0} \int_{-\infty}^{+\infty} f_\epsilon(x) \varphi(x) \, dx[/tex]
(the limit with the hat is the new kind of limit, the other limit is the ordinary kind you learn in calculus)
(and, for the record, the integral on the left is not the same integral you learned in calculus either)