I've got another one for you.

I ran across this theorem in my Field Theory text. The theorem is this:

Given a sufficiently smooth function f(x) we have that:

$\displaystyle \lim_{x \to \infty}f(x) + \lim_{x \to - \infty}f(x) = \lim_{\epsilon \to 0^+} \epsilon \int_{-\infty}^{\infty} dx f(x) e^{-\epsilon |x|}$

The term "sufficiently smooth" does not require that f(x) is a $\displaystyle C^{\infty}$ function, but as most wavefunctions in Physics are $\displaystyle C^{\infty}$ I would be satisfied with a proof of this case. (Note: Obviously we are concerned only with f(x) that have limits that existat $\displaystyle \pm \infty$.)

The only f(x) I can figure out how to integrate (as a test of the theorem) is a Gaussian. All the other functions I know of with the required properties at infinity is discontinous at at least one point on the real line, so it doesn't qualify.

-Dan