Problem 1.

In this problem, we will compute the following definite integral:
$$\int_0^{2 \pi} \frac{\sin n \theta}{\sin \theta} d \theta$$
for $n=1,2,3, \ldots$

1. (5 points) Convert the integral (1) into the contour integral of a complex integral over the positively oriented unit circle $C$.
2. (5 points) By computing the contour integral, evaluate (1) when $n=2$. That is, evaluate
$$\int_0^{2 \pi} \frac{\sin 2 \theta}{\sin \theta} d \theta$$
3. (5 points) By computing the contour integral, evaluate (1) for $n=1,2,3, \ldots$, ,

Take a parametrization for C
\begin{aligned} & C: z=e^{i \theta} \quad(0 \leq \theta \leq 2 \pi) \ & \left(\begin{array}{l} \sin \theta=\left(e^{i \theta}-e^{-i \theta}\right) / 2 i=\left(z-z^{-1}\right) / 2 i \ \sin n \theta=\left(e^{i n \theta}-e^{-i n \theta}\right) / 2 i=\left(z^n-z^{-n}\right) / 2 i \ d z=i \cdot e^{i \theta} d \theta \Rightarrow d \theta=d z / i z \end{array}\right. \ & \int_0^{2 \pi} \frac{\sin n \theta}{\sin \theta} d \theta=\int_c \frac{z^n-z^{-n}}{z-z^{-1}} \cdot \frac{d z}{i z} \ & =\frac{1}{i} \cdot \int_c \frac{z^n}{z^n} \cdot \frac{z^n-z^{-n}}{z-z^{-1}} \cdot \frac{d z}{z} \ & =\frac{1}{i} \cdot \int_c \frac{z^{2 n}-1}{z^n\left(z^2-1\right)} d z \ & =\frac{1}{i} \cdot \int_c \frac{1}{z^n} \cdot\left(1+z^2+\cdots+z^{2 n-2}\right) d z \ & =\frac{1}{i} \cdot \int_C\left(z^{-n}+z^{-n+2}+\cdots+z^{n-2}\right) d z \ & \end{aligned}
\begin{aligned} & \Leftrightarrow z^{2 n}-1=\left(z^2-1\right)\left(z^{2 n-2}+\cdots+z^2+1\right) \ & \Rightarrow \frac{z^{2 n}-1}{z^n\left(z^2-1\right)}=\frac{1}{z^n} \cdot\left(1+z^2+\cdots+z^{2 n-2}\right) \end{aligned}
$z= \pm 1$ are removable singular points
We can extend $\frac{z^{2 n}-1}{z^n\left(z^2-1\right)}$ at $z= \pm 1$ by using
$$\frac{1}{z^n} \cdot\left(1+z^2+\cdots+z^{2 n-2}\right)$$

Note that $z^{-n}+z^{-n+2}+\cdots+z^{n-2}$

contains $z^{-1} \quad$ if $n=1.3,5, \cdots$ (odd)
$$\Rightarrow \operatorname{Res}_{z=0}\left(z^{-n}+z^{-n+2}+\cdots+z^{n-2}\right)=1$$

does not contain $z^{-1}$ if $n=2.46, \cdots$ (even)
$$\Rightarrow \operatorname{Res}_{z=0}\left(z^{-n}+z^{-n+2}+\cdots+z^{n-2}\right)=0$$
By Cauchy’s residue theorem,
$$\int_0^{2 \pi} \frac{\sin n \theta}{\sin \theta} d \theta= \begin{cases}\frac{1}{i} \cdot 2 \pi i \cdot 1=2 \pi & \text { if } n \text { is odd } \ \frac{1}{i} \cdot 2 \pi i \cdot 0=0 & \text { if } n \text { is even }\end{cases}$$

Problem 2.

Let $C$ be the positively oriented unit circle centered at the origin. Let
$$f(z)=z^{2020}-5 z^{115}+z^{100}-2 z$$
Evaluate the following contour integral
$$\int_C \frac{f^{\prime}(z)}{f(z)} d z$$

Let $f_1(z):=-5 z^{115}$ and $f_2(z):=z^{2020}+z^{100}-2 z$
We then have $f(z)=f_1(z)+f_2(z)$
Observe

$\left|f_1(z)\right|=5 \cdot|z|^{115}=5$ on $C$

$\left|f_2(z)\right| \leq|z|^{2020}+|z|^{100}+2|z|=4$ on $C$
$\Rightarrow|f(z)| \geq\left|f_1(z)\right|-\left|f_2(z)\right|=1>0$ at each $p t z \in C$
By argument principle (or its proof), we have
$$\int_C \frac{f^{\prime}(z)}{f(z)} d z=2 \pi i(z-P)$$
where

$z$ : = the number of zeros of $f$ inside $C$ counting multip.

$p:=$ poles
Since $f$ is entire, $P=0$
So, it suffices to compute $Z$ to evaluate the integral.
To compute Z, we’ll apply Rouche’s theorem.
$$f(z)=f_1(z)+f_2(z)$$

$f_1 \& f_2$ are entire

$\otimes \Rightarrow\left|f_1(z)\right|>\left|f_2(z)\right|$ at each pt $z \in C$
By Rouche’s theorem,
\begin{aligned} & z=\text { ( the number of zeros of } f_1(z)=-5 z^{347} \text { inside } C \text { ) } \ & =115 \ & \int_C \frac{f^{\prime}(z)}{f(z)} d z=2 \pi i \cdot 115=230 \pi i \ & \end{aligned}

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# MATH0013 Analysis 3: Complex Analysis

 Year: 2023-2024 Code: MATH0013 Level: 5 (UG) Normal student group(s): UG: Year 2 Mathematics degrees Value: 15 credits (= 7.5 ECTS credits) Term: 1 Assessment: The weighting of the module is 80% exam, 9% written coursework, 5% test, 4% quizzes, 2% participation.In the participation mark, 1% will be awarded for attending 3 or more tutorials and 1% will be awarded for giving a presentation in a tutorial.(The final exam for MATH0013 will take place during the main exam period, April/May 2023) Normal Pre-requisites: MATH0003 (MATH0004 recommended) Instructor: Prof A Sobolev

## Course Description and Objectives

This is a course on complex functions. The treatment is rigorous. Starting from complex numbers, we study some of the most celebrated theorems in analysis, for example, Cauchy’s theorem and Cauchy’s integral formulae, the theorem of residues and Laurent’s theorem. The course lends itself to various applications to real analysis, for example, evaluation of definite integrals and finding the number of zeros of a complex polynomial in a region.

## Recommended Texts

Relevant books are:

1. Stein, E. and Shakarchi, R., Complex analysis (Princeton Lectures in Analysis)
2. Priestley, H. A., Introduction to complex analysis (Oxford)
3. Stewart, I. and Tall, D., Complex Analysis (CUP)
4. Churchill, J. W. and Brown, R., Complex Variables and Applications (McGraw-Hill Higher Education, 8th edition)

## Detailed Syllabus

### The complex numbers and topology in $\mathbb{C}$

Review of complex numbers. Neighbourhoods, open and closed sets. Convergence of sequences. The Riemann sphere.

### Functions on the complex plane

Continuous, holomorphic functions, Cauchy-Riemann equations, and power series. Harmonic functions.

### Cauchy’s Theorem and its applications

Integration along curves. Goursat’s theorem, Cauchy’s theorem in the disc, Cauchy’s integral formulas. Taylor’s theorem, Laurent’s theorem, Liouville’s theorem

Fourier analysis代写

## 离散数学代写

Partial Differential Equations代写可以参考一份偏微分方程midterm答案解析