Yes, the saddle point approximation is useful in a couple of limits, but these limits are basically just the trivial limits because the integrand is almost exactly Gaussian (or a pair of Gaussians) in these limits.
Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average:
$$
I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)}
$$
for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...
Hi everyone,
in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like:
I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)}
where a,b,c are...
Hi wc2351,
it just so happens that I just solved this problem for the class I am TAing.
You have the right idea for the proof, but let me lay down the final steps in order for the benefit of anyone who might read it. The key is that the two-electrons Schrödinger equation for the orbital...