1. Oct 18, 2006

Let's suppose we have an integral:

$$\int_{-\infty}^{\infty}dXe^{-sF(X)}=g(s)$$

where $$X=(x_1,x_2,x_3,....,x_n)$$

Then for big s we could use "Saddle point method " and all that stuff well, we still have the problem in finding a "meaning" for g(s) for every positive s, the idea i had (of course i know it has ben named before ) is appart from making the expansion near the "stationary points" so $$dF(X)=0$$ is to evaluate in some "wise" points satisfying some other constraint, i.e $$dF(X)+F(x)=0$$ or something similar, the idea came to me when examining "Gaussian Cuadrature" where you evalute your function plus a weight function at certain points that are the roots of Laguerre/Hermite?¿ (i can't remember ) my idea is if something similar can be made to give meaning to the inegral g(s) for s>0 BUT not big (let's say s=1) or for example imposing an extra "multiplier" so the integral becomes:

$$\int_{-\infty}^{\infty}dXe^{-sF(X)-s\lambda g(X)}=g(s)$$