There is an extra 1/s in your final formula. Let \(\displaystyle{f{{\left({t}\right)}}}={a}{e}^{{-{b}{t}}}{u}{\left({t}\right)}\) with \(\displaystyle{a},{b}\in{R}\) (I'm assuming your u(t) is the Heaviside step function). Then

PSK(Lf)(s)=\int_0^{\infty} f(t)e^{-st} dt=\int_0^{\int} ae^{-bt}e^{-st}u(t)dtask

\(\displaystyle={a}{\int_{{0}}^{{\infty}}}{e}^{{-{\left({b}+{s}\right)}{t}}}{\left.{d}{t}\right.}={\frac{{{a}}}{{{s}+{b}}}}\)

The region of convergence is \(\displaystyle{s}\in{C}:{R}{e}{\left({s}\right)}\succ{b}\)

PSK(Lf)(s)=\int_0^{\infty} f(t)e^{-st} dt=\int_0^{\int} ae^{-bt}e^{-st}u(t)dtask

\(\displaystyle={a}{\int_{{0}}^{{\infty}}}{e}^{{-{\left({b}+{s}\right)}{t}}}{\left.{d}{t}\right.}={\frac{{{a}}}{{{s}+{b}}}}\)

The region of convergence is \(\displaystyle{s}\in{C}:{R}{e}{\left({s}\right)}\succ{b}\)