For simplicity, let us consider a function $f$ holomorphic on a domain $D \subseteq \mathbb{C}^2$. We may therefore write $f$ as a sum of power series $$f(z) = \sum_{\nu_1 \nu_2 =0}^{\infty} c_{\nu_1 \nu_2}(z_1 - w_1)^{\nu_1} (z_2 - w_2)^{\nu_2}.$$

We should also note that the domain of convergence of a power series in $>1$ complex variable is not a polydisk as one would expect, but rather a logarithmically convex Reinhardt domain.

In the case of one complex variable, the standard way of determining the radius of convergence is by means of the ratio test. Is there an analogous "standard approach" for determining the domain of convergence of a power series of two complex variables?

For example, the power series $$\frac{1}{1 - z_1z_2} = \sum_{\nu=0}^{\infty} z_1^{\nu} z_2^{\nu}$$ converges for $\left| z_1 z_2 \right| <1$, while the series $$\frac{z_1}{(1-z_1)(1-z_2)} = \sum_{\left| k \right| =0}^{\infty} z_1^{k_1 + 1} z_2^{k_2}$$ converges in the bidisk $\{ \left| z_1 \right|< 1, \left| z_2 \right| < 1\}$, completed by the complex line $\{ z_1 =0 \}$.

[Reference: Shabat's Introduction to Complex analysis].


The usual Cauchy-Hadamard formula has a generalization to several variables. The numbers $r_1,\ldots,r_n$ are called conjugate radii of convergence if the series converges in the open polydisk $B(r_1,\ldots,r_n)$ and diverges in $\{ z:|z_j|>r_j, 1\leq j\leq n\}$. Then we have the formula $$\limsup_{|k|\to\infty}\left(|c_{k_1\ldots,k_n}|r_1^{k_1}\ldots r_n^{k_n}\right)^{|k|}=1,$$ where $|k|=k_1+\ldots+k_n.$

Reference: B. A. Fuks, Theory of analytic functions of several variables, vol. I, Chap. I, sect 3, Theorem 3.7.

  • $\begingroup$ Thank you very much. +1 for also including a reference, cheers. $\endgroup$
    – AmorFati
    Nov 9 '17 at 21:54
  • $\begingroup$ Looking at Shabat's text again, he also includes this result, which leads me to a question that was rather implicit in the original question. That is, are (any of) the elementary series tests that we were all taught in first year valid for functions of several complex variables? $\endgroup$
    – AmorFati
    Nov 9 '17 at 22:02
  • $\begingroup$ In $\{z : \lvert z_j\rvert > r_j, 1 \le j \le n\}$, is the implicit quantification "for all $j$" or "for some $j$"? $\endgroup$
    – LSpice
    Jan 12 at 17:02
  • $\begingroup$ For some $j$ but with the understanding that it can accidentally converge at some points but not on an open set in the complement of $B$. $\endgroup$ Jan 12 at 21:04

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