Power series in several variables

Summary

A power series in several variables is an infinite sum $\sum_\alpha c_\alpha (z - a)^\alpha$ of monomials in $z_1 - a_1, \dots, z_n - a_n$, indexed by multi-indices $\alpha$. It extends the one-variable power series. A function on an open set of $\C^n$ is holomorphic exactly when it is locally the sum of a convergent power series. The domain of convergence of such a series is the interior of the set where it converges. Unlike in one variable, it is not always a polydisc; it is a logarithmically convex complete Reinhardt domain.