What is the Laplace Transform of $\displaystyle f(x)=e^{bx}.sin ax$
hmmmm.... as I recall from signal and systems, Laplace transformation doesn't exist on periodical signals like u have there ...
there is Laplace transformation of $\displaystyle \displaystyle sin (\omega_0 t) u(t) $ because with step function $\displaystyle u(t) $ u put to zero your signal for all t<0, But with like this one u have (periodical) it doesn't converge
P.S. perhaps only if u use unilateral Laplace transformation (but still I'm spinning a lot even that way) which limits go from (-0) to infinity... but as I said i think there's no point going there because it just start going in circles (at least I do)
Edit : without your "constants" (a) and (b) and using unilateral formula (don't know if you are bounded to use bilateral) u get this :
$\displaystyle \displaystyle \frac {1}{S^2-2S+2} $ that's for $\displaystyle ROC : Re(S) >1 $ (region of convergence )
without (a), or should i say a=1 u'll get :
$\displaystyle \displaystyle \frac {1}{(b-S)^2+1} $ that's for $\displaystyle ROC :Re(b)< Re(S) $
yes, always u can use table of Laplace transformations, but on the exams i don't think anywhere would someone allow you to use table of Laplace transformations... or that's just me
anyway using unilateral u can get to the solution, but using bilateral u can't
solution should be something like:
$\displaystyle \displaystyle \frac {a}{(S-b)^2+a^2} $ with $\displaystyle ROC: Re(S)>Re(a) $