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I would like a HINT for this:

Exhibit a two variable power series whose convergence domain is the unit ball $\{(z,w):|z|^2+|w|^2 < 1\}$.

($z$ and $w$ are complex numbers.) I think that it cannot be of the form $\sum P(z,w)^n$ where $P(z,w)$ is a polynomial. But I'm out of ideas.

Thank you.

Weltschmerz
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2 Answers2

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Hints:

  1. Consider the series $\sum_{k=0}^\infty (z^2 + e^{ik} w^2)^k$.
  2. If $|z^2| + |w^2| = \rho< 1$ then $|z^2 + e^{ik} w^2| \leq \rho$ for all $k$, and thus the series converges. If $|z^2| + |w^2| > 1$ then $|z^2 + e^{ik} w^2| > 1$ for infinitely many $k$, and the series diverges.
Yury
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According to Lebl's book "Tasty Bits of Several Complex Variables", power series in several complex variables are considered for the absolute convergence. The reason behind that is non-existence of a canonical way of ordering terms in the power series of several variables.

So I will provide my answer according to following definitions:

Definition 1: A series $\sum c_{jk}z_1^jz_2^k$ is said to be absolutely convergent if the series $\sum \vert c_{jk}z_1^jz_2^k\vert$ convergent.

Definition 2: For a power series $\sum c_{jk}z_1^jz_2^k$, the domain of convergence is the interior of the subset of $\mathbb{C}^2$ where the power series is absolutely convergent.

Now answer to the question:

Start with a real power series of single variable $\sum a_nx^n$ whose radius of convergence is $1$ and replace $x^n$ with $(z_1^2e^{if_1(n)}+z_2^2e^{if_2(n)})^n$ where $f_1(n)$ and $f_2(n)$ any sequence of real numbers (or simply replace $x$ with $z_1^2+z_2^2$ and you get your answer.

But how? (I will show when $x$ is replaced with $z_1^2+z_2^2$ )

We have $$\sum_{n=0}^{\infty} a_n(z_1^2+z_2^2)^n=\sum_{n=0}^{\infty} \sum_{j=0}^n a_n \binom{n}{j}z_1^{2j}z_2^{2n-2j}\quad .$$ Now if you revisit definition 1 we have the following:

\begin{align} \sum_{n=0}^{\infty} \sum_{j=0}^n \vert a_n \binom{n}{j}z_1^{2j}z^{2n-2j}\vert=&\sum_{n=0}^{\infty} \sum_{j=0}^n \vert a_n\vert \binom{n}{j}\vert z_1 \vert^{2j}\vert z_2\vert^{2n-2j}\\=&\sum_{n=0}^{\infty} a_n(\vert z_1\vert^2+\vert z_2\vert^2)^n\quad . \end{align}

Now w.l.o.g. we can assume the real power series we started converges only when $\vert x \vert<1$. We can assume this because in definition 2 we only care about the interior. Then we see that $$\sum_{n=0}^{\infty} a_n(z_1^2+z_2^2)^n$$ converges absolutely if and only if $\vert z_1\vert^2+\vert z_2\vert^2<1$. So the series $$\sum_{n=0}^{\infty} a_n(z_1^2+z_2^2)^n$$ has desired domain of convergence.

For example: $$\sum_{n=1}^{\infty} (-1)^{n+1}\frac{(z_1^2+z_2^2)^n}{n}$$ is a specific answer to the question since we use Taylor's series of $\ln(1+x)$ here.

Platonicsolids
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