Is the Saddle Point Expansion Valid for Finite Values of x?

[Kevin_spencer2]

To evaluate the integral

$$\int_{-\infty}^{\infty}dt e^{xf(t)}$$ whenever x is 'big' (tending to infinity) we use the saddle point expansion so:

$$\int_{-\infty}^{\infty}dt e^{xf(t)}\sim g(x)\sum_{n=0}^{\infty}a_{n}x^{-n}$$

Of course the expansion above is just valid for x---> infinite, but what would happen if i put x=1 and hence i must find the sum for the a(n):

$$\sum_{n=0}^{\infty}a(n) = S$$ will at least S exist in the sense of a 'Borel summable' series to calculate the integral for x=1,2,3,4,...

 

[fresh_42]

A large $x$ is the reason we do all this. However, in this treatise you can find the general case
http://bolvan.ph.utexas.edu/~vadim/Classes/2011f/saddle.pdf