Reading Notes, Theorems

Convergence of power series: Cauchy-Hadamard

The domain of convergence of a general holomorphic function can be quite arbitrary—in one (complex) dimension any domain is the domain of convergence for some holomorphic function, and in higher dimensions domains of holomorphy may be characterized using certain notions of convexity: holomorphic convexity, which involves the propagation of maximal-principle-type estimates, or pseudoconvexity, which is similar but uses the larger class of plurisubharmonic functions instead of holomorphic functions

The domain of convergence of functions defined by power series, however, are much more restricted and exhibit more symmetry: in one dimension, the domain of convergence of a power series f(z) = \sum_{n=0}^\infty c_n z^n is a disc of radius R satisfying \dfrac 1R = \limsup_{n \to \infty} \left( |c_n|^{1/n} \right). This we may prove by noting that, for |z| > R with R as above, the terms of the series do not converge to zero, and hence the series cannot converge; for |z| < R, the series will converge by comparison with a geometric series. This is the Cauchy-Hadamard theorem. (On the boundary of the disc of convergence the behavior of the power series is more subtle—we can extract some information using results such as Abel’s theorem.)

There is an analogue in higher dimensions, which states that (using multi-index notation) a power series f(z) = \sum_\alpha c_\alpha z^\alpha converges in a polydisc with polyradius \rho iff \limsup_{|\alpha| \to \infty} \sqrt[|\alpha|]{|c_\alpha|\rho^\alpha} \leq 1; the proof of this is exactly analogous to the one given above. Note that the union of all such polydiscs may not itself be a polydisc–e.g. the domain of convergence of the power series \sum_{k=0}^\infty z_1^k z_2^k is precisely the set \{|z_1z_2|| < 1\}, which is not a polydisc.

Nevertheless these domains of convergence still have a certain sort of radial symmetry—they belong to the more general sort of domains known as a complete (a.k.a. logarithmically-convex) Reinhardt domains. These are domains U for which (z_1, \dots, z_n) \in U implies (e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n) \in U for all \theta_1, \dots, \theta_n \in \mathbb{R} (this is the “Reinhardt” part) and (w_1, \dots, w_n) \in U whenever |w_i| \leq |z_i| (this is the complete, or log-convex part.) Note that one-dimensional complete Reinhardt domains are precisely discs centered at the origin.

We can think of the Cauchy-Hadamard condition in several variables as giving a constraint on the polyradius \rho, so that the domain of convergence \Omega is the union of all polydiscs with polyradius satisfying the given constraint or, equivalently, the logarithmic image \mathrm{Log}(\Omega) of the domain of convergence is the set of all polyradii satisfying the given constraint. \mathrm{Log}(\Omega) is convex (and closed to the right) but \Omega need not be an orthant, as would be the case if \Omega were a polydisc.