Gameline Complex Analysis, P. 265 #8 is like this,
Show that every conformal self-map of the complex plane $ \mathbb C$ is linear.
Hint: The isolated singularity of $f(z)$ must be the simple pole.
First of all, how do I argue that the singularity is not essential or removable? Second of all, how do I argue it is a pole and is simple?
I can see there is a singularity at $ \infty$ because function is not really defined there. Some hints please!
Addendum Now I can see that the singularity can not be removable because of the Liouville's Theorem. If the singularity at $\infty$ were removable then that would make function bounded and hence analyticity in the entire complex plane tells function is constant but which is impossible because function is bijective.