Evaluating this integral (Principal value)

1. Oct 27, 2006

Let's suppose we wish to calculate:

$$PV \int_{0}^{\infty}dx\frac{e^{-sx}}{1-x}$$ (1)

I don't know how to do it, my idea is take the identity:

$$\frac{1}{1-x}=1+x+x^{2}+x^{3}+............ |x|<1$$

$$\frac{1}{1-x}=-x^{-1}-x^{-2}-x^{-3}+x^{3}+............ |x|>1$$

So (1) using Cauchy's principal value, is the same as to calculate with a positive small epsilon:

$$\int_{0}^{1-\epsilon}dxx^{k} e^{-sx}$$

$$\int_{1+\epsilon}^{\infty}dxx^{-k} e^{-sx}$$

and calculate term by term integration to get the Principal value of (1)