Region of convergence

One more:
after doing Laplace transform for this:

[tex] f(t) = e^{(7+5j)t}u(t-1) [/tex]

where u(t) = 1 for t >= 0 and 0 otherwise;
so here's what I have:

[tex]L[f(t)] = \frac {e^{-(s-7-5j)}}{s-7-5j} [/tex]

so, my reasoning was that it would converge if Re > 7 because that's the value for which exponential would converge. But why exactly do we not care about Im? I know that by Euler's formula, it [tex]e^{jw}[/tex] would just be oscillating but don't I need a condition for denominator of L[f(t)]?
Thanks for your time and explanation.

 

Thanks for reply.

yeah, i know, that's what I was asking about: do I need to state a condition for the denominator to exclude a case where it is = 0?

edit: what are the cases when Re >= or <= some value?
let's say I have [tex] L[f(t)] = \frac{1}{s-a}[/tex]
so, region of convergence would be Re >= a or Re > a?

thanks again.