When I saw Wikipedia's notation for the inverse Laplace transform, I became curious if there was a reason behind it.
Is there a reason why Wikipedia writes the inverse Laplace transform as this $$\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,$$ instead of this?$$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}e^{st}F(s)\,ds,$$
(Is there something wrong with putting an infinity in the bounds?)
In the case of integrals over the reals, it's common to write $$ \lim_{t\to\infty}\int_a^t f(x)\,dx=\int_a^\infty f(x)\,dx $$ and similarly for the lower bound. However, $$ \int_{-\infty}^\infty f(x)\,dx $$ means $$ \lim_{\substack{s\to-\infty\\t\to\infty}}\int_s^t f(x)\,dx $$ Note that it can happen that this limit doesn't exist, while $$ \lim_{t\to\infty}\int_{-t}^t f(x)\,dx $$ does.
In the case you're showing there's also a different problem: what “path” should be followed for going from $\gamma-i\infty$ to $\gamma+i\infty$? In the complex plane there's no way to define $-\infty$ distinct from $\infty$. I don't want to say that one cannot give a meaning to the notation you're proposing; it's actually not used, because it's too ambiguous.