The Alternate form of the Dirac Delta Function

[WilcoRogers]

Hello,

I am trying to show that:
\delta(x) = \lim_{\epsilon \to 0} \frac{\sin(\frac{x}{\epsilon})}{\pi x}
Is a viable representation of the dirac delta function. More specifically, it has to satisfy:
<br /> \int_{-\infty}^{\infty} \delta(x) f(x) dx = f(0)<br />

I know that the integral of sin(x)/x over the reals is \pi, and so far as I can tell, it doesn't depend on epsilon. What I've tried so far is integration by parts, which leads me to:
<br /> f(x) - \int_{-\infty}^{\infty} f&#039;(x) dx<br />

Which isn't really getting me somewhere, and the limit drops off due to the integral not caring what epsilon is. Is there another way of approaching this? Or am I on the right track, I just can't pull out an f(0) from this.

Any help is appreciated,

Thanks.

 

[obafgkmrns]

Try substituting z = x/epsilon.

 

[vela]

Are you familiar with complex analysis and the residue theorem?