How to obtain a bound on Gamma function on a vertical strip?

Question

I am interested in obtaining a bound for the Gamma function $\Gamma(z)$ on a vertical strip. Let $z = x + iy$ and suppose $A \leq x \leq B$ then I want to obtain a bound on $|\Gamma (x + iy)|$. I have read before that it decays very quickly on vertical strips, but I was wondering which property/equation I can use to obtain a bound for it. Thank you very much.

Answer 1

For $\sigma > 0$ $$\Gamma(\sigma+i\omega) = \int_0^\infty x^{\sigma+i\omega-1} e^{-x}dx =\int_{-\infty}^\infty e^{-u(\sigma +i\omega)} e^{-e^{-u}}du=\mathcal{F}[g_\sigma(u)](\omega)$$ where $\mathcal{F}$ is the Fourier transform and $g_\sigma(u) =e^{-u\sigma} e^{-e^{-u}}$ is a Schwartz function. Therefore

$$|(i\omega)^k \Gamma(\sigma+i\omega)|=|\mathcal{F}[g_\sigma^{(k)}(u)](\omega)| \le \|g_\sigma^{(k)}\|_{L^1}, \qquad \forall k,\Gamma(\sigma+i\omega) = o(\omega^{-k})$$ This extends to every $\sigma$ using $\Gamma(s)= \frac{\Gamma(s+1)}{s}$.

You can obtain a much better bound after showing the reflection formula $$\Gamma(s)\Gamma(1-s) = \frac{\pi}{\sin(\pi s)}$$