[Math] Bounded, non-constant, analytic function on C\[-1,1]

Question

Find an explicit analytic function on $\mathbb{C}\setminus[-1,1]$ which is bounded and non-constant.

Suggestions on how to approach this problem?

Answer 1

Try to find a holomorphic function from your domain into the unit disk. Proceed in steps and start with $$ T(z) = \frac{z-1}{z+1} $$ which maps $\mathbb{C}\setminus[-1,1]$ into the complex plane without the negative real axis. Continue with a mapping into the right half-plane, and you are almost done.

Spoiler:

$$ f(z) = \dfrac{\sqrt{\frac{z-1}{z+1}} - 1}{\sqrt{\frac{z-1}{z+1}} + 1}$$