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# 数学代写|多复变函数论代考Multivariable Complex Analysis代写|MATHS7101 Multiple power series and multicircular domains

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## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Multiple power series and multicircular domains

In the following we will study sets of convergence of power series and of more general Laurent series
$$\sum_{\alpha \in \mathbb{Z}^n} c_\alpha z^\alpha=\sum_{\alpha_1 \in \mathbb{Z}, \ldots, \alpha_n \in \mathbb{Z}} c_{\alpha_1 \ldots \alpha_n} z_1^{\alpha_1} \ldots z_n^{\alpha_n} .$$
In order to avoid problems with the order of the terms, we only consider absolute convergence here.

Definition 2.3.1. Let $A$ be the set of those points $z \in \mathbb{C}^n$ where the Laurent series (2.3.1) [or power series (2.3.2)] is absolutely convergent. The interior $A^0$ of $A$ will be called the domain of (absolute) convergence of the series.

In the case $n=1$ the domain of convergence is an open annulus or disc (or empty). For general $n$, our first observation is that the absolute convergence of a Laurent series (2.3.1) at a point $z$ implies its absolute convergence at every point $z^{\prime}$ with $\left|z_j^{\prime}\right|=\left|z_j\right|, \forall j$. Indeed, one will have $\left|c_\alpha\left(z^{\prime}\right)^\alpha\right|=\left|c_\alpha z^\alpha\right|, \forall \alpha$. It is convenient to give a name to the corresponding sets of points:
Definition 2.3.2. $E \subset \mathbb{C}^n$ is called a multicircular set (or Reinhardt set) if
$$a=\left(a_1, \ldots, a_n\right) \in E \quad \text { implies } \quad a^{\prime}=\left(e^{i \theta_1} a_1, \ldots, e^{i \theta_n} a_n\right) \in E$$
for all real $\theta_1, \ldots, \theta_n$. A multicircular domain or Reinhardt domain is an open multicircular set.
Multicircular sets are conveniently represented by their “trace” in the space $\mathbb{R}_{+}^n$ “of absolute values”, in which all coordinates are nonnegative. Cf. Figure 1.5, where the multicircular domain $D=\Delta\left(0,0 ; 2, \frac{1}{2}\right) \cup \Delta\left(0,0 ; \frac{1}{2}, 2\right)$ in $\mathbb{C}^2$ is represented by its trace.

## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Convergence domains of power series and analytic continuation

Let $B$ denote the set of those points $z \in \mathbb{C}^n$ at which the terms $c_\alpha z^\alpha, \alpha \in \mathbb{Z}{\geq 0}^n$ of the power series (2.3.2) form a bounded sequence: $$B=\left{z \in \mathbb{C}^n:\left|c\alpha z^\alpha\right| \leq M=M(z)<+\infty, \quad \forall \alpha \in \mathbb{Z}{\geq 0}^n\right} .$$ The set $B$ is clearly multicircular and it also has a certain convexity property: Lemma 2.4.1. The trace of $B$ is logarithmically convex. Proof. Let $r^{\prime} \geq 0$ and $r^{\prime \prime} \geq 0$ be any two points in $\operatorname{tr} B$. Then there is a constant $M$ [for example, $\left.M=\max \left{M\left(r^{\prime}\right), M\left(r^{\prime \prime}\right)\right}\right]$ such that $$\left|c\alpha\right|\left(r_1^{\prime}\right)^{\alpha_1} \ldots\left(r_n^{\prime}\right)^{\alpha_n} \leq M, \quad\left|c_\alpha\right|\left(r_1^{\prime \prime}\right)^{\alpha_1} \ldots\left(r_n^{\prime \prime}\right)^{\alpha_n} \leq M, \quad \forall \alpha \in \mathbb{Z}{\geq 0}^n .$$ It follows that for any $r=\left(r_1, \ldots, r_n\right)$ with components of the form $r_j=\left(r_j^{\prime}\right)^{1-\lambda}\left(r_j^{\prime \prime}\right)^\lambda[$ with $\lambda \in[0,1]$ independent of $j]$ and for all $\alpha^{\prime} s$, $$\left|c\alpha\right| r_1^{\alpha_1} \ldots r_n^{\alpha_n}=\left{\left|c_\alpha\right|\left(r_1^{\prime}\right)^{\alpha_1} \ldots\left(r_n^{\prime}\right)^{\alpha_n}\right}^{1-\lambda}\left{\left|c_\alpha\right|\left(r_1^{\prime \prime}\right)^{\alpha_1} \ldots\left(r_n^{\prime \prime}\right)^{\alpha_n}\right}^\lambda \leq M .$$
Thus $r \in B$ and hence $\operatorname{tr} B$ is logarithmically convex [Definition 2.2.3].

# 多复变函数论代考

## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Multiple power series and multicircular domains

$$\sum_{\alpha \in \mathbb{Z}^n} c_\alpha z^\alpha=\sum_{\alpha_1 \in \mathbb{Z}2 \ldots, \alpha_n \in \mathbb{Z}} c{\alpha_1 \ldots \alpha_n} z_1^{\alpha_1} \ldots z_n^{\alpha_n}$$

$$a=\left(a_1, \ldots, a_n\right) \in E \quad \text { implies } \quad a^{\prime}=\left(e^{i \theta_1} a_1, \ldots, e^{i \theta_n} a_n\right) \in E$$

## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Convergence domains of power series and analytic continuation

\left 缺少或无法识别的分隔符

$$|c \alpha|\left(r_1^{\prime}\right)^{\alpha_1} \ldots\left(r_n^{\prime}\right)^{\alpha_n} \leq M, \quad\left|c_\alpha\right|\left(r_1^{\prime \prime}\right)^{\alpha_1} \ldots\left(r_n^{\prime \prime}\right)^{\alpha_n} \leq M, \quad \forall \alpha \in \mathbb{Z} \geq 0^n .$$

\left 缺少或无法识别的分隔符

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