[seanhunter]

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).

[1] https://mathworld.wolfram.com/GaussianIntegral.html

[niklasbuschmann]

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.

[srean]

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

[chermi]

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?

[srean]

It is Laplace's method of steepest decent. When the same method is used to approximate probability density function, for example the posterior probability density, it's called Laplace's approximation of the density.

The wikipedia link would have made things quite clear :)

[dawnofdusk]

They are the same for physicists who analytically continue everything. Steepest descent is technically for integrals in the complex plane, and Laplace's method is for integrating over the real line.

[constantcrying]

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.