I've recently come across a result I've been trying to generalize.
Say that $\phi(\sigma \pm iy) < Ce^{\frac{\pi}{2}|y|}$ in the strip $a < \sigma < b$
then then the following integral is convergent for all positive $x$ minus zero.
$$g(x) = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \Gamma(s) \phi(s) x^{-s}\,ds$$
I've been able to prove (given $\phi(0)$ is known) that $\lim_{x\to 0^+} g(x) = \phi(0)$ in some special cases (i.e., if $a = -\infty$, or if $\phi$ satisfies some similar bound conditions to the left of $a$) .
I believe this result should work for any strip. But I do not know how to show this without doing contour integration (which is how I showed it for the special case). Therefore my main question is, if $\phi$ satisfies the bounds in some strip and $\phi(0)$ is known, then does:
$$\phi(0) = \lim_{x \to 0^+}\frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \Gamma(s) \phi(s) x^{-s}\,ds$$
Any help would be greatly appreciated. Any literature on the subject, or even hints will be greatly appreciated. Thanks a lot.