# 复分析代考_Complex function代考_MATH3711

## 复分析代考_Complex function代考_Runge’s Theorem

A rational function is, by definition, a quotient of polynomials. A function from $\widehat{\mathbb{C}}$ to $\widehat{\mathbb{C}}$ is rational if and only if it is meromorphic on all of $\widehat{\mathbb{C}}$ (Theorem 4.7.7). A rational function has finitely many zeros and finitely many poles. The polynomials are those rational functions with pole only at $\infty$. Let $K \subseteq$ $\mathbb{C}$ be compact and let $f: K \rightarrow \mathbb{C}$ be a given function on $K$. Under what conditions is $f$ the uniform limit of rational functions with poles in $\widehat{\mathbb{C}} \backslash K$ ? There are some obvious necessary conditions:
(1) $f$ must be continuous on $K$;
(2) $f$ must be holomorphic on $\stackrel{\circ}{K}$, the interior of $K$.
It is a striking result of Mergelyan that, in case $\widehat{\mathbb{C}} \backslash K$ has finitely many connected components, then these conditions are also sufficient. We shall prove Mergelyan’s theorem in the next section. In the present section we shall establish a slightly weaker result-known as Runge’s theorem-that is considerably easier to prove.

Theorem 12.1.1 (Runge). Let $K \subseteq \mathbb{C}$ be compact. Let $f$ be holomorphic on a neighborhood of $K$. Suppose that $P$ is a subset of $\widehat{\mathbb{C}} \backslash K$ containing one point from each connected component of $\widehat{\mathbb{C}} \backslash K$. Then for any $\epsilon>0$ there is a rational function $r(z)$ with poles in $P$ such that
$$\sup _{z \in K}|f(z)-r(z)|<\epsilon$$

Corollary 12.1.2. Let $K \subseteq \mathbb{C}$ be compact with $\widehat{\mathbb{C}} \backslash K$ connected. Let $f$ be holomorphic on a neighborhood of $K$. Then for any $\epsilon>0$ there is a holomorphic polynomial $p(z)$ such that
$$\sup _K|p(z)-f(z)|<\epsilon .$$
The corollary follows from the theorem by taking $P={\infty}$.
We shall prove Theorem 12.1.1 later in this section, but first we shall give some applications and explain the idea of the proof.

## 复分析代考_Complex function代考_Mergelyan’s Theorem

It is easy to see that the proof of Runge’s theorem in Section $12.1$ genuinely relied on the hypothesis that $f$ is holomorphic in a neighborhood of $K$. After all, we needed some place to put the contour $\gamma$. This leaves open the question whether conditions (1) and (2) at the beginning of Section $12.1$ are sufficient for rational approximation on $K$. When $\widehat{\mathbb{C}} \backslash K$ has finitely many components, the conditions are sufficient: This is the content of Mergelyan’s theorem which we prove in this section. It is startling that a period of sixty-seven years elapsed between the appearance of Runge’s theorem (1885) and that of Mergelyan’s theorem (1952), for Mergelyan’s proof involves no fundamentally new ideas. The proof even uses Runge’s theorem. The sixtyseven year gap is even more surprising if one examines the literature and sees the large number of papers written on this subject during those years. The explanation is probably that people thought that Mergelyan’s theorem was too good to be true. It took the world by surprise and superseded a huge number of very technical partial results that had been proved by others.
We shall concentrate first on proving the simplest form of Mergelyan’s theorem, using elementary and self-contained methods (the proof that we give here follows the steps in [RUD2]). Afterwards, in this section and in Section $12.3$, we shall explore further results of the same type.

Theorem 12.2.1 (Mergelyan). Let $K \subseteq \mathbb{C}$ be compact and assume that $\widehat{\mathbb{C}} \backslash K$ is connected. Let $f \in C(K)$ be holomorphic on the interior $\stackrel{\circ}{K}$ of $K$. Then for any $\epsilon>0$ there is a holomorphic polynomial $p(z)$ such that
$$\sup _{z \in K}|p(z)-f(z)|<\epsilon .$$
The proof is most readily grasped if it is broken into several lemmas: In this way, the main ideas can be identified.

# 复分析代考

## 复分析代考_Complex function代考_Runge’s Theorem

(1) $f$ 必须是连续的 $K$;
(2) $f$ 必须是全纯的 $K$ ，的内部 $K$.
Mergelyan 的一个惊人结果是，如果 $\widehat{\mathbb{C}} \backslash K$ 有有限多个连通分量，则这些条件也充分。我们将在下一节证 明 Mergelyan 定理。在本节中，我们将建立一个稍微弱一点的结果一一称为龙格定理一一它更容易证 明。

$$\sup _{z \in K}|f(z)-r(z)|<\epsilon$$ 推论 12.1.2。让 $K \subseteq \mathbb{C}$ 与 $\widehat{\mathbb{C}} \backslash K$ 连接的。让 $f$ 在的邻域上是全纯的 $K$. 然后对于任何 $\epsilon>0$ 存在一个全纯多 项式 $p(z)$ 这样
$$\sup _K|p(z)-f(z)|<\epsilon .$$

## 复分析代考_Complex function代考_Mergelyan’s Theorem

$$\sup _{z \in K}|p(z)-f(z)|<\epsilon .$$

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