# 数学代写|复分析作业代写Complex function代考|MATH2521

#### Doug I. Jones

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## 数学代写|复分析作业代写Complex function代考|Examples of Laurent Expansions

In this section we consider several examples which illustrate techniques for calculating Laurent expansions. First, some terminology: When $f$ has a pole at $P$, it is customary to call the negative power part of the Laurent expansion of $f$ around $P$ the principal part of $f$ at $P$. That is, if
$$f(z)=\sum_{j=-k}^{\infty} a_{j}(z-P)^{j}$$
for $z$ near $P$, then the principal part of $f$ at $P$ is
$$\sum_{j=-k}^{-1} a_{j}(z-P)^{j} \text {. }$$
Next, we give an algorithm for calculating the coefficients of the Laurent expansion:

Proposition 4.4.1. Let $f$ be holomorphic on $D(P, r) \backslash{P}$ and suppose that $f$ has a pole of order $k$ at $P$. Then the Laurent series coefficients $a_{j}$ of $f$ expanded about the point $P$, for $j=-k,-k+1,-k+2, \ldots$, are given by the formula
$$a_{j}=\left.\frac{1}{(k+j) !}\left(\frac{\partial}{\partial z}\right)^{k+j}\left((z-P)^{k} \cdot f\right)\right|_{z=P} .$$
Proof. This is just a direct calculation with the Laurent expansion of $f$ and is left as an exercise (Exercise 29) for you.

## 数学代写|复分析作业代写Complex function代考|The Calculus of Residues

In the previous section we focused special attention on functions that were holomorphic on punctured discs, that is, on sets of the form $D(P, r) \backslash{P}$. The terminology for this situation was to say that $f: D(P, r) \backslash{P} \rightarrow$ $\mathbb{C}$ had an isolated singularity at $P$. It turns out to be useful, especially in evaluating various types of integrals, to consider functions which have more than one “singularity” in this same informal sense. More precisely, we want to consider the following general question: Suppose that $f: U \backslash$ $\left{P_{1}, P_{2}, \ldots, P_{n}\right} \rightarrow \mathbb{C}$ is a holomorphic function on an open set $U \subseteq \mathbb{C}$ with finitely many distinct points $P_{1}, P_{2}, \ldots, P_{n}$ removed. Suppose further that $\gamma:[0,1] \rightarrow U \backslash\left{P_{1}, P_{2}, \ldots, P_{n}\right}$ is a piecewise $C^{1}$ closed curve. Then how is $\oint_{\gamma} f$ related to the behavior of $f$ near the points $P_{1}, P_{2}, \ldots, P_{n}$ ?

The first step is naturally to restrict our attention to open sets $U$ for which $\oint_{\gamma} f$ is necessarily 0 if $P_{1}, P_{2}, \ldots P_{n}$ are removable singularities of $f$. Without this restriction we cannot expect our question to have any reasonable answer. We thus introduce the following definition:

Definition 4.5.1. An open set $U \subseteq \mathbb{C}$ is holomorphically simply connected (abbreviated h.s.c.) if $U$ is connected and if, for each holomorphic function $f: U \rightarrow \mathbb{C}$, there is a holomorphic function $F: U \rightarrow \mathbb{C}$ such that $F^{\prime} \equiv f$.
The connectedness of $U$ is assumed just for convenience. The important part of the definition is the existence of holomorphic antiderivatives $F$ for each holomorphic $f$ on $U$. The following statement is a consequence of our earlier work on complex line integrals.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Examples of Laurent Expansions

$$f(z)=\sum_{j=-k}^{\infty} a_{j}(z-P)^{j}$$

$$\sum_{j=-k}^{-1} a_{j}(z-P)^{j}$$

$$a_{j}=\left.\frac{1}{(k+j) !}\left(\frac{\partial}{\partial z}\right)^{k+j}\left((z-P)^{k} \cdot f\right)\right|_{z=P} \text {. }$$

## 数学代写|复分析作业代写Complex function代考|The Calculus of Residues

left{P_{1}, P_{2}, Vdots, P_{n}}right } \rightarrow \mathbb{C} 是开集上的全纯函数 $U \subseteq \mathbb{C}$ 具有 有限多个不同点 $P_{1}, P_{2}, \ldots, P_{n}$ 删除。进一步假设 线。那么怎么样 $\oint_{\gamma} f$ 与行为有关 $f$ 靠近点 $P_{1}, P_{2}, \ldots, P_{n}$ ?

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## MATLAB代写

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