# Perturbation theory.

1. Feb 2, 2008

### mhill

Hi, i am stuck at this problem , let be the divergent quantity

$$m= clog(\epsilon) +a_{0}+a_{1}g\epsilon ^{-1}+a_{2}g\epsilon ^{-2} +a_{3}g\epsilon ^{-3}+.........+$$

where epsilon tends to 0 and g is just some coupling constant and c ,a_n are real numbers.

then i use the Borel transform of the function $$F(t)= \sum_{n=0}^{\infty}a_{n} \frac{t^{n}}{n!}$$ in this case

$$m= clog(\epsilon)+ \int_{0}^{\infty}dtF(t/\epsilon)e^{-t}$$

my question is, does this last expression have only 2 divergent quantities ? , mainly the one due to log(e) and the second involving the poles of $$F(t/\epsilon)$$