Convergence domain: $\{(z,w):|z|^2+|w|^2 < 1\}$

Question

I would like a HINT for this:

Exhibit a two variable power series whose convergence domain is the unit ball $\{(z,w):|z|^2+|w|^2 < 1\}$.

($z$ and $w$ are complex numbers.) I think that it cannot be of the form $\sum P(z,w)^n$ where $P(z,w)$ is a polynomial. But I'm out of ideas.

Thank you.

Answer 1

Hints:

1. Consider the series $\sum_{k=0}^\infty (z^2 + e^{ik} w^2)^k$.
2. If $|z^2| + |w^2| = \rho< 1$ then $|z^2 + e^{ik} w^2| \leq \rho$ for all $k$, and thus the series converges. If $|z^2| + |w^2| > 1$ then $|z^2 + e^{ik} w^2| > 1$ for infinitely many $k$, and the series diverges.