这是一次UCL伦敦大学学院图论与组合(MATH0029)课程的代写成功案例
这门课是一门本科生的复分析的入门课。最开始就是从复数的构造开始讲起。首先介绍了实分析中一些标准的拓扑概念，如领域（neighborhood）和紧致性（compactness），以及数列的收敛性。还讲解了实分析中收敛性的等价性。因为在复分析中，我们研究的函数大部分是整函数（entire function），整函数实际上存在于黎曼球（Riemann sphere）上，它可以无限远地延拓到无穷远点。然后讲解了在复平面上的函数，首先是连续性，然后是我们最重要的性质，即全纯函数（holomorphic function）。接下来讲解了全纯函数的一个刻画，即柯西-黎曼方程（Cauchy-Riemann equations），并讲解了全纯函数在局部的展开，即幂级数（power series）。还讲解了全纯函数的实部和虚部都是调和函数，需要注意的是柯西-黎曼方程以及全纯函数展开的表达式，包括调和函数都是全纯函数在不同角度下的刻画。然后讲解了柯西积分公式（Cauchy’s integral formula）。从最简单的情况开始讲证明，然后我们可以发现柯西积分公式对于一般的闭曲线都成立。通过柯西积分公式，我们可以得到一种迭代的过程，用于证明所谓的泰勒展开定理。此外，还证明了非亚纯函数的所有洛朗级数展开，此时我们从调和函数的角度可以得到诸如刘维尔定理（Liouville’s theorem）和以及代数基本定理（Fundamental theorem of algebra）。这些都可以被视为调和函数性质的概述，或者是全纯函数的简要介绍。最后，讲解了亚纯函数（meromorphic function）以及亚纯函数相关的内容，例如留数定理（residue theorem）和最大模原理（maximum modulus principle）。这些都是复分析里面最基础的内容，并没有涉及到像黎曼映照定理的证明也没有用复分析的方法证明迪利克莱定理（Dirichlet’s theorem）或者讨论黎曼曲面以及多复变的知识的内容。
In this problem, we will compute the following definite integral:
$$
\int_0^{2 \pi} \frac{\sin n \theta}{\sin \theta} d \theta
$$
for $n=1,2,3, \ldots$
- (5 points) Convert the integral (1) into the contour integral of a complex integral over the positively oriented unit circle $C$.
- (5 points) By computing the contour integral, evaluate (1) when $n=2$. That is, evaluate
$$
\int_0^{2 \pi} \frac{\sin 2 \theta}{\sin \theta} d \theta
$$ - (5 points) By computing the contour integral, evaluate (1) for $n=1,2,3, \ldots$, ,
这个题的处理思路就是我们把这个函数用复变量去参数化，我们知道欧拉公式可以把z表示成指数形式，那么我们的函数就可以表示为e的i乘以theta减去e的-i乘以theta，这样子的话就可以将整个表达式用指数形式表示完之后我们对它进行一些化简。最后我们可以利用留数定理进行计算。
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,
$$
\int_0^{2 \pi} \frac{\sin n \theta}{\sin \theta} d \theta= \begin{cases}\frac{1}{i} \cdot 2 \pi i \cdot 1=2 \pi & \text { if } n \text { is odd } \ \frac{1}{i} \cdot 2 \pi i \cdot 0=0 & \text { if } n \text { is even }\end{cases}
$$
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
$$
\int_C \frac{f^{\prime}(z)}{f(z)} d z
$$
因为我们要处理的这个对象比较复杂，我们希望找到另外两个多项式。其中一个多项式的模长可以控制，而另一个多项式计算根比较容易。然后我们可以利用所谓的幅角原理（argument principle）和罗赫定理（Rouché’s theorem）。通过应用围道积分定理（Cauchy’s integral formula），我们可以计算所需多项式的零点。这个积分就等于计算根比较容易的多项式的积分。
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:
- Stein, E. and Shakarchi, R., Complex analysis (Princeton Lectures in Analysis)
- Priestley, H. A., Introduction to complex analysis (Oxford)
- Stewart, I. and Tall, D., Complex Analysis (CUP)
- 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.
Elementary functions $e^z, \sin z, \cos z, \log z, z^\alpha$ and conformal mapping. Linear fractional transformations.
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
微分几何代写
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
