Personal revision on MA244 Analysis III, learn from tutorial sheet and past papers.
This repository will mainly focus on two parts, support class questions and past papers. I will provide the pdf made by myself so there will not be any problems about license.
- MA244-Analysis-III-Revision
The most important part is of course the lecture notes, feel free to download the 2022 version.
Notes-Analysis-III-JLR-2022.pdf
Below are all the support class questions I have done, and it is divided into weekly sections, feel free to download the complete version created by myself.
In week 2's support class questions, the most important concept is Riemann integrable.
- Let
$f:\left[a,b\right]\to\mathbb{R}$ be a bounded function. Then$f$ is integrable if and only if for every$\varepsilon > 0$ , there exists a partition$P$ of$\left[a,b\right]$ such that$$U(f,P) - L(f,P) < \varepsilon.$$ - Intuitively,
$$L(f,P)\leq \int_{a}^{b}f\leq U(f,P).$$
Don't forget refinement as well!!!
- A partition
$Q = \set{J_{1},...,J_{l}}$ of$\left[a,b\right]$ is a refinement of a partition$P = \set{I_{1},...,I_{n}}$ if every interval$I_{k}$ in$P$ is the union of one or more intervals$J_{k}$ from the partition$Q$ .
In week 3's support class questions, there are some theorems that is worth investigating.
- If
$f:[a,b]\to\mathbb{R}$ is continuous, then if$u$ is a number between$f(a)$ and$f(b)$ , that is$$\min (f(a),f(b))\leq u\leq\max((f(a),f(b)),$$ then there exists a$c\in[a,b]$ , such that$$f(c) = u.$$
- A function
$f:\Omega\subset\mathbb{R}\to\mathbb{R}$ is uniformly continuous if for every$\varepsilon>0$ , there exists a$\delta = \delta(\varepsilon)$ , such that$$\left|x-y\right|<\delta\implies\left|f(x)-f(y)\right|<\varepsilon.$$
Something really obvious but needs to be remembered:
- Consider two elements
$a,b$ , we have$$\left|\left|a\right|-\left|b\right|\right|\leq\left|a-b\right|.$$
- Consider two elements
$a,b$ , we have$$\left|a+b\right|\leq\left|a\right|+\left|b\right|.$$
In week 4's support class questions, the topic is more about improper integrals.
- Let
$f:(a,b]\to\mathbb{R}$ be a Riemann integrable function for every$[c,b]$ with$a < c$ . Then the improper integral of$f$ on$[a,b]$ is defined as$$\int_{a}^{b}f(x)\mathrm{d}x = \lim_{\varepsilon\to0^{+}}\int_{a+\varepsilon}^{b}f(x)\mathrm{d}x.$$
- For
$x \geq 1$ , we have$$\log (x) \leq x-1.$$
In week 5's support class questions, there are two main focuses: pointwise convergence and uniform convergence.
- Let
$\left(f_{n}\right)$ be a sequence of functions with$f_{n}$ :$\Omega \to \mathbb{R}$ . We say that$(f_{n})$ or$f_{n}$ converges pointwise to $f:\Omega\to\mathbb{R}% if and only if for every$x\in\Omega$ , we have$$\lim_{n\to\infty}f_{n}(x) = f(x).$$
- Let
$f_{n}:\Omega\to\mathbb{R}$ be a sequence of functions. We say that$(f_{n})$ converges uniformly to$f:\Omega\to\mathbb{R}$ if and only if for every$\varepsilon>0$ there exists$N(\varepsilon)$ such that$$\left|f_{n}(x) - f(x)\right|<\varepsilon$$ for every$x\in\Omega$ and for all$n>N(\varepsilon).$ - Its notation is
$$f_{n}\rightrightarrows f.$$ - Also, we can have a simplier version:
$$f_{n}\rightrightarrows f \leftrightarrow \forall\varepsilon > 0,\exists N(\varepsilon), s.t. \sup_{x\in\Omega}\left|f_{n}-f\right|<\varepsilon , \forall n> N(\varepsilon).$$
- Uniform convergence implies pointwise convergence. The converse if false and the example is given in Page 22 in Notes and 2(a) in Support class problems.
In week 6's support class questions, we need to remember the following theorems we learnt in 1st year.
- We say a function f has a limit at infinity, if there exists a real number
$L$ such that for all$ε>0$ , there exists$N>0$ such that$$\left|f(x)−L\right|<ε$$ for all$x>N$ . In that case, we write$$\lim_{x\to\infty}f(x)=L.$$
- If
$f$ is a continuous function on a closed interval$\left[a,b\right]$ and differentiable on the open interval$(a,b)$ , then there exists a point$c\in(a,b)$ such that$$f'(c) = \frac{f(b)-f(a)}{b-a}.$$
- Remember
$$\sum_{n=1}^{\infty}\frac{1}{n^{p}}$$ converges when$p>1$ .
Suppose that
-
$\left|f_{n}(x)\right|\leq M_{n}$ , for all$n\geq 1$ and$x\in A.$ -
$\displaystyle\sum_{n=1}^{\infty} M_{n}$ converges.
Then the series
- Assume that
$f_{n}$ converges uniformly to$f$ on$C$ and that each$f_{n}$ is uniformly continuous on$C$ , then$f$ is uniformly continuous on$C$ .
The proof is easy.
- A sequence
$(f_{n})$ of functions in$\Omega$ is called uniformly Cauchy if and only if for every$\varepsilon > 0$ , there exists$N(\varepsilon)$ such that$\left||f_{n}-f_{m}\right||_{\infty} < \varepsilon$ for all$n,m>N(\varepsilon)$ . - A sequence
$(f_{n})$ is uniformly covergent if and only if it is uniformly Cauchy.
- Let
$(f_{n})$ be a sequence of$C^{1}$ functions on$[a,b]$ . Assume$f_{n}\to f$ in the pointwise sense and$f_{n}'\rightrightarrows g$ . Then$f$ is$C^{1}$ and$g = f'$ or$f_{n}'\rightrightarrows f'$ .
In week 7's support class, these two theroems are still need to be mainly focused on.
Suppose that
-
$\left|f_{n}(x)\right|\leq M_{n}$ , for all$n\geq 1$ and$x\in A.$ -
$\displaystyle\sum_{n=1}^{\infty} M_{n}$ converges.
Then the series
- Let
$(f_{n})$ be a sequence of$C^{1}$ functions on$[a,b]$ . Assume$f_{n}\to f$ in the pointwise sense and$f_{n}'\rightrightarrows g$ . Then$f$ is$C^{1}$ and$g = f'$ or$f_{n}'\rightrightarrows f'$ .
In week 8's support class, we are entering Complex Analysis and the main focus is Cauchy-Riemann equation and Series convergence.
- Assume that
$f(z) = u(x,y)+iv(x,y)$ for$z = x+ iy$ . Then the Cauchy-Riemann equations are given by$$u_{x} = v_{y},\qquad u_{y} = -v_{x}.$$
- Given a sequence
$(a_{n})$ , there exists$R\in\left[0,\infty\right]$ called the radius of convergence such that$$\sum_{n=0}^{\infty}a_{n}z^{n}$$ converges for all$\left|z\right| < R$ and diverges for$\left|z\right| > R$ . - There is a formula for finding such
$R$ ,$$R = \frac{1}{\displaystyle\lim_{n\to\infty}\sup\left|a_{n}\right|^{\frac{1}{n}}}.$$ - Ratio test (first version): Consider
$\displaystyle\sum_{n=0}^{\infty}a_{n}$ and assume that$a_{n}\ne 0$ for all$n$ . Then 1. If$\lim\sup\frac{|a_{n}+1|}{|a_{n}|} < 1$ , then$\displaystyle\sum_{n=0}^{\infty}a_{n}$ is convergent; 2. If$\frac{|a_{n+1}|}{|a_{n}|}\geq 1$ for all$n>N$ , then$\displaystyle\sum_{n=0}^{\infty}a_{n}$ is divergent. - Ratio test (second version): Let
$a_{n}\ne 0$ for all$n\geq N$ , and assume that$\displaystyle\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_{n}|}$ exists. Then$\displaystyle\sum_{n=0}^{\infty}a_{n}z^{n}$ has radius of convergence$$R = \lim_{n\to\infty}\frac{|a_{n}|}{|a_{n+1}|}.$$
In week 9's support class, the main focus is still about Cauchy-Riemann equations and its applications. Apart from that, there are still some more theorems to check.
- Let
$f:\Omega\to\mathbb{C}$ be an analytic function, with$\Omega$ an open, simply connected domain. Let$\gamma$ be a$C^{1}$ closed curve in$\Omega$ . Then$$\int_{\gamma}f(z)\mathrm{d}z = 0.$$
- Techniques: Since we are in complex analysis area, remember the parametrisation of
$z$ always sticks to$z = x+iy$ . For example, when you are trying to parametrise a line$y = x$ , we rewrite$z$ as$e^{i\theta}$ . Note when$t\in[0,1]$ ,$\left|z\right|$ should move from 1 to 0 (assume counter-clockwise). Hence the first part of the parametrisation should be$(1-t)$ (the choice of$\theta$ would not affect$\left|z\right|$ as$\left|e^{i\theta}\right| = 1$ .). - For
$y = x$ , the line makes$\frac{\pi}{4}$ to the$x$ -axis, so$\theta = \frac{\pi}{4}$ in this case. - Therefore, the final parametrisation is
$z = (1-t)e^{i\frac{\pi}{4}}$ .
- Techniques: This is fairly straightforward, the parametrisation should be
$$z = Re^{it},\quad t\in[0,2\pi].$$ - If the question is asking to calculate
$$I = \frac{1}{2\pi i}\int_{\partial B_{r}(0)} f(z)\mathrm{d}z,$$ where$f:z\to\displaystyle \sum_{n=0}^{\infty}a_{n}z^{n},$ subsutitute the parametrisation and do the normal integral. Note there might be a involve of Kroneck Delta.
- The Kroneck Delta is defined as when
$i\ne j$ ,$$\delta_{ij} = 1, $$ when$i = j$ ,$$\delta_{ij} = 0.$$
In week 10's support class, the main focus is still Cauchy's formula. However there are more to look at.
- Let
$\gamma:\left[a,b\right]\to\mathbb{C}$ be a positively oriented simple closed$C^{1}$ curve. Assume that$f$ is analytic in$\gamma$ and on the interior of$\gamma$ ,$I(\gamma)$ . Then$$f(z) = \frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-z}\mathrm{d}w,\qquad \forall z\in I(\gamma).$$
The following is only a small part of the principle:
- Let
$f$ be an analytic function in$B_{r}(z)$ for some$r>0$ and$z\in\mathbb{C}$ , and we parametrise$z = a+re^{it}$ , then$$\int_{0}^{2\pi} \left|f(z+re^{i\theta})\right|\mathrm{d}\theta \leq \max_{\theta}\left|f(z+re^{i\theta})\right|\int_{0}^{2\pi}\mathrm{d}\theta.$$ - The reason for the above inequality is although
$\displaystyle\max_{\theta}\left|f(z+re^{i\theta})\right|$ involves$\theta$ which should be integrated, it is a constant so we can put it outside the integral.
- Let
$f$ be an analytic function on$B_{R}(a)$ for$a\in\mathbb{C}$ ,$R>0$ . Then there exists unique constants$c_{n}, n\in\mathbb{N}$ such that$$f(z) = \sum_{n=0}^{\infty}c_{n}(z-a)^{n}\qquad\forall z\in B_{R}(a).$$ Moreover, the coefficients$c_{n}$ are given by$$c_{n} = \frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{(w-a)^{n+1}}\mathrm{d}w = \frac{f^{(n)}(a)}{n!}.$$
- Let
$f:\mathbb{C}\to\mathbb{C}$ be an analytic, bounded function. Then$f$ is constant.