I have some questions regarding the construction of the gamma function below. First, why is $|t^{z-1}|=t^{\Re z-1}$? Next, why is $|f_z(t)|\le C_z e^{-t/2}$ for $t\ge 1$, and how can the precise value of $C_z$ be found as it is indicated? Finally, why do we need two bounds for $|f_z(t)|$, to prove integrability? I would greatly appreciate any explanations as I've been stuck here for a while.
(1): It is easy to show that $|e^z| = e^{\Re z}$ by the series definition, from which $|t^{z-1}| = t^{\Re z - 1}$ immediately follows.
(2): You can use differential calculus to find the maximum value of $C_z$ by Fermat's Theorem, but there is no reason to do so.
(3): Since $t \geq 1$, $$ |f_z(t)| = |t^{z-1} e^{-t}| \leq |t^{z-1}| = t^{\Re z - 1} $$ $$ |f_z(t)| = |t^{z-1} e^{-t}| = |t^{z-1} e^{-t/2}| e^{-t/2} \leq C_z e^{-t/2} $$
(4): The two bounds indicate that $|f_z(t)|$ is bounded, and from continuity it is integrable.