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.