In about two weeks, I'm going to be giving a presentation on the complex-valued Gamma function $\Gamma(z)$. By definition, I know that $\Gamma(z)= \int_0^\infty e^{-t}t^{z-1}dt.$
Now if I let $z=x+iy$, how does the following hold? $|\Gamma(x+iy)| \leq \Gamma(x).$
It might actually be something quite simple, but here's what I attempted: $|\Gamma(x+iy)|= |\int_0^\infty e^{-t}t^{z-1}dt|$ $=|\int_0^\infty e^{-t}t^{x+iy-1}dt|$ $=|\int_0^\infty e^{-t}t^{x-1}t^{iy}dt|$ $\leq|\int_0^\infty e^{-t}t^{x-1}dt|$ $=\int_0^\infty e^{-t}t^{x-1 }dt$ $=\Gamma(x),$
where $t \gg 1$. I hope this was the correct procedure.