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]. 



Practically calculating the domain of a power series for function of several complex variables
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 does not converge on any open subset of
$\{ z:|z\_j|>r\_j, 1\leq j\leq n\}$. 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.


