How to split complex fractions like $\frac{e^{-as}}{s(1-e^{-as})}$ using partial fraction technique?

Question

$$\frac{e^{-as}}{s(1-e^{-as})}=\frac{A}{s}+\frac{B}{1-e^{-as}}$$

Multiply the whole thing by $s(1-e^{-as})$ $$e^{-as}=A(1-e^{-as})+Bs$$

I distribute and try to find A and B by matching coefficients.

I get $A=1, -1$ and $B=0$. Now what?

Turns out by a little trial and error, I get: $$-\frac{1}{s}+\frac{1}{s(1-e^{-as})}$$

Answer 1

Add and substract $1$ upstairs to have:

$$\frac{e^{as}}{s (1-e^{as})} = \frac{1 - 1 +e^{as}}{s (1-e^{as})} = -\frac{1}{s} + \frac{1}{s (1-e^{as})}$$