# 数学代写|复分析作业代写Complex function代考|MATH3401

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

One of the most classical and fascinating applications of the calculus of residues is the calculation of definite (usually improper) real integrals. It is an over-simplification to call these calculations, taken together, a “technique”: It is more like a collection of techniques. We can present only several instances of the method and ask you to do lots of problems to sharpen your skills.

The main interest of the method which we are about to present is that it allows us to calculate many improper integrals which are not tractable by ordinary techniques of calculus. However, we shall begin (for simplicity) with an example which could in principle be done with calculus by using partial fractions.
EXAMPLE 4.6.1. To evaluate
$$\int_{-\infty}^{\infty} \frac{1}{1+x^{4}} d x$$
we “complexify” the integrand to $f(z)=1 /\left(1+z^{4}\right)$ and consider the integral
$$\oint_{\gamma_{R}} \frac{1}{1+z^{4}} d x$$
See Figure 4.4. Note that since $\lim {a \rightarrow \infty} \int{0}^{a} \frac{1}{1+x^{4}} d x$ and $\lim {b \rightarrow \infty} \int{-b}^{0} \frac{1}{1+x^{4}} d x$ exist separately, the improper integral exists and there is no harm in evaluating the integral as $\int_{-R}^{R} \frac{1}{1+x^{4}} d x$, with $R \rightarrow \infty$.

Now part of the game here is to choose the right piecewise $C^{1}$ curve or “contour” $\gamma_{R}$. The appropriateness of our choice is justified (after the fact) by the calculation which we are about to do. Assume that $R>1$. Define
\begin{aligned} &\gamma_{R}^{1}(t)=t+i 0 \quad \text { if } \quad-R \leq t \leq R \ &\gamma_{R}^{2}(t)=R e^{i t} \quad \text { if } \quad 0 \leq t \leq \pi \end{aligned}

## 数学代写|复分析作业代写Complex function代考|Meromorphic Functions and Singularities at Infinity

We have considered carefully functions which are holomorphic on sets of the form $D(P, r) \backslash{P}$ or, more generally, of the form $U \backslash{P}$, where $U$ is an open set in $\mathbb{C}$ and $P \in U$. Sometimes it is important to consider the possibility that a function could be “singular” at more than just one point. The appropriate precise definition requires a little preliminary consideration of what kinds of sets might be appropriate as “sets of singularities”. Recall from Section 3.6:

Definition 4.7.1. A set $S$ in $\mathbb{C}$ is discrete if and only if for each $z \in S$ there is a positive number $r$ (depending on $S$ and on $z$ ) such that
$$S \cap D(z, r)={z} .$$
We also say in this circumstance that $S$ consists of isolated points.
Now fix an open set $U$; we next define the central concept of meromorphic function on $U$.

Definition 4.7.2. A meromorphic function $f$ on $U$ with singular set $S$ is a function $f: U \backslash S \rightarrow \mathbb{C}$ such that
(a) the set $S$ is closed in $U$ and is discrete,
(b) the function $f$ is holomorphic on $U \backslash S$ (note that $U \backslash S$ is necessarily open in $\mathbb{C}$ ),
(c) for each $z \in S$ and $r>0$ such that $D(z, r) \subseteq U$ and $S \cap D(z, r)=$ ${z}$, the function $\left.f\right|_{D(z, r) \backslash{z}}$ has a (finite order) pole at $z$.

# 复分析代写

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

$$\int_{-\infty}^{\infty} \frac{1}{1+x^{4}} d x$$

$$\oint_{\gamma_{R}} \frac{1}{1+z^{4}} d x$$

$\gamma_{R}^{1}(t)=t+i 0 \quad$ if $\quad-R \leq t \leq R \quad \gamma_{R}^{2}(t)=R e^{i t} \quad$ if $\quad 0 \leq t \leq \pi$

## 数学代写|复分析作业代写Complex function代考|Meromorphic Functions and Singularities at Infinity

$$S \cap D(z, r)=z .$$

(a) 集合 $S$ 封闭在 $U$ 并且是离散的，
(b) 函数 $f$ 是全纯的 $U \backslash S$ (注意 $U \backslash S$ 一定是开放的 $\mathbb{C}$ )，
(c) 对于每个 $z \in S$ 和 $r>0$ 这样 $D(z, r) \subseteq U$ 和 $S \cap D(z, r)=z$ ，功能 $\left.f\right|_{D(z, r) \backslash z}$ 有一个 (有限阶) 极点 $z$.

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