Comment by seanhunter
1 day ago
I thought when I first saw the title that it was going to be about the Gaussian integral[1] which has to be one of the coolest results in all of maths.
That is, the integral from - to + infinity of e^(-x^2) dx = sqrt(pi).
I remember being given this as an exercise and just being totally shocked by how beautiful it was as a result (when I eventually managed to work out how to evaluate it).
Gaussian integrals are also pretty much the basis of quantum field theory in the path integral formalism, where Isserlis's theorem is the analog to Wick's theorem in the operator formalism.
Indeed.
It's the gateway drug to Laplace's method (Laplace approximation), mean field theory, perturbation theory, ... QFT.
https://en.m.wikipedia.org/wiki/Laplace%27s_method
Maybe I should just read the wiki you linked, but I guess I'm confused on how this is different than steepest descent? I'm a physicist by training so maybe we just call it something different?
2 replies →
There is a relationship here, in the case of Gauß-Hermite Integration, where the weight function is exactly e^(-x^2) the weights have to add up sqrt(pi), because the integral is exact for the constant 1 polynomial.