Domain of convergence of a power series

Remarks

If $z \in D$, there is a point $w \in S$ with $\abs{w_j} > \abs{z_j}$ for all $j$. Then Abel's lemma gives convergence on the polydisc $\set{\abs{w'_j} < \abs{w_j}}$, which contains $z$ and every $w'$ with $\abs{w'_j} \le \abs{z_j}$. This gives the first and third parts. For the second part, suppose the terms are bounded by $M$ at $z$ and at $w$, and let $0 \le t \le 1$. If $\abs{u_j} = \abs{z_j}^{1-t} \abs{w_j}^{t}$ for all $j$, then $\abs{c_\alpha u^\alpha} = \abs{c_\alpha z^\alpha}^{1-t} \abs{c_\alpha w^\alpha}^{t} \le M$. So the set of logarithms of absolute values is convex. Example: the series $\sum_{k \ge 0} (z_1 z_2)^k$ has domain of convergence $\set{\abs{z_1 z_2} < 1}$, whose image under $z \mapsto (\log\abs{z_1}, \log\abs{z_2})$ is the half-plane $x_1 + x_2 < 0$. This domain is unbounded and is not a polydisc, which never happens for $n = 1$, where the domain of convergence is a disc or the whole plane. Conversely, every logarithmically convex complete Reinhardt domain with centre $0$ is the domain of convergence of some power series.

Going deeper

For a student

This result explains why Reinhardt domains and logarithmic convexity appear so early in the theory. It is a first example of the convexity conditions that characterise domains of holomorphy. For instance, a holomorphic function on a complete Reinhardt domain $D$ with centre $0$ has a power series at $0$ that converges on all of $D$. Hence it converges on the smallest logarithmically convex complete Reinhardt domain containing $D$. So the holomorphic functions on a complete Reinhardt domain that is not logarithmically convex, such as a union of two crossed polydiscs, all extend to a larger domain.