Representation of Dirac-delta function

[quantumkiko]

Homework Statement

Show that $$\delta_\epsilon(x) = \frac{\epsilon}{\pi (x^2 + \epsilon^2)}$$ is a representation of the Dirac delta function.

Homework Equations

I already know how the function can satisfy the first two requirements of being a dirac delta function, namely

$$\lim_{\substack{\epsilon \to 0}} \delta_\epsilon (x) = 0$$,
$$\int_{-\infty}^{\infty} \delta_\epsilon (x) dx = 1$$.

The third requirement is

$$\lim_{\substack{\epsilon \to 0}} \int_{-\infty}^{\infty} f(x) \delta_\epsilon (x) dx = f(0)$$.

The Attempt at a Solution

Following the previous examples of the book, it seems that the solution should be like this..

$$\frac{\epsilon}{\pi} \int_{-\infty}^{\infty} \frac{f(x)}{x^2 + \epsilon^2} dx = \frac{\epsilon}{\pi} \, f(c\epsilon) \int_{-\infty}^{\infty} \frac{1}{x^2 + \epsilon^2} dx = \frac{\epsilon}{\pi} \frac{\pi}{\epsilon} f(c\epsilon) = f(c\epsilon)$$

where c varies from negative infinity to positive infinity. Now as we take the limit of epsilon to zero, we get f(0) which is the answer. But I don't get how the 2nd step (the one where the f(ce) came out of the integral) was done using the mean value theorem. I've looked up the integral forms of the mean value theorem but I still couldn't figure out how the 2nd line is done.

 

[hunt_mat]

How about applying the Cauchy residue theorem? This should yield a function of $$\varepsilon$$ which you can take the limit without a problem.