Chapter 14.11, Problem 3MP

Liouville’s theorem: Suppose $f(z)$ is analytic for all $z$ (except $\infty$), and bounded [that is, $|f(z)|\le M\text{ for all }z\text{ and some }M].$ Prove that $f(z)$ is a constant. Hints: If $f^{\prime }(z)=0,$ then $f(z)=$ const. To show this, write $f^{\prime }(z)$ as in Problem 3.21 where $C$ is a circle of radius $R$ and center z, that is, $w=z+R{e}^{i\theta }.$ Show that and let $R\to \infty$.