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