Analysis 2 - Concise Notes 1

MATH50001

Term 1 Content

PDFs

Problem Sheets - Term 1

Problem Sheets - Term 2

Table of contents

Term I
1 Differentiation in Higher Dimensions
2 Metric and Topological Spaces
Term II
Holomorphic Functions
Cauchy’s Integral Formula
Applications of Cauchy’s integral formula
Meromorphic Functions
Harmonic Functions

Colour Code - Definition are green in these notes, Consequences are red and Causes are blue

Content from MATH40002 assumed to be known.

Term I

1 Differentiation in Higher Dimensions

1.1 Euclidean Spaces

1.1.1 Preliminaries

Definition - Modulus Function

\[|x| := \begin{cases} x, & x\geq 0 \\ -x, & x \leq 0 \end{cases}\]

Having the following properties:

$\forall x \in \mathbb{R}, \lvert x \rvert \geq 0, \lvert x \rvert = 0 \iff x = 0$
$\forall x, y \in \mathbb{R}, \lvert xy \rvert = \lvert x \rvert\lvert y \rvert$
$\forall x, y \in \mathbb{R}, \lvert x+y \rvert \leq \lvert x \rvert + \lvert y \rvert$ (Triangle inequality)

1.1.2 Euclidean space of dim. n

Define - Euclidean Space of dim. $n, \mathbb{R}^{n}$

Defined as the set of ordered $n$-tuples $(x^{1},\dots,x^{n})$, s.t $\forall i,\ x^{i} \in \mathbb{R}$

$\mathbb{R}^{n} $ a vector space.

Define - Inner Product $ < \cdot,\cdot > : \mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}$

\[\langle(x^{1},x^{2},\dots,x^{n}),(y^{1},y^{2},\dots,y^{n})\rangle = \sum_{i=1}^n x^{i}y^{i}\]

Define - Norm/Lengths, $\mid\mid\cdot\mid\mid: \mathbb{R}^{n} \to \mathbb{R}$

\[||x|| = \sqrt{<x,x>}\]

Having the following properties:

$\forall x \in \mathbb{R}^{n}, \lvert\lvert x\rvert\rvert \geq 0, \lvert\lvert x\rvert\rvert \iff x = \vec{0}$
$\forall \lambda x \in \mathbb{R}, x\in \mathbb{R}^{n} \lvert\lvert \lambda x\rvert\rvert = \lvert\lambda\rvert\lvert\lvert x\rvert\rvert$
$\forall x, y \in \mathbb{R}^{n}, \lvert\lvert x+y\rvert\rvert \leq \lvert\lvert x\rvert\rvert + \lvert\lvert y\rvert\rvert$ (Triangle inequality)

Definition - Cauchy-Schwartz Inequality

\[|\langle x, y \rangle| \leq ||x||\cdot||y||\]

1.1.3 Convergence of Sequences in Euclidean Spaces

Definition - Sequence in $\mathbb{R}^{n}$

An infinite ordered list, $x_{0}, x_{1},\dots,$ s.t $x_{i} \in \mathbb{R}^{n} \text{ } \forall i$

Denoted $(x_{i})_{i\geq 1}$ or $(x_{i})_{i\in \mathbb{N}}$

Definition 1.1 - Convergence

A seq. $(x_{i}) \in \mathbb{R}^{n}$ converges to $x \in \mathbb{R}^{n}$ if $\forall \epsilon > 0, \exists N \in \mathbb{N}\text{ s.t } \forall i \geq \mathbb{N}, ||x_{i}-x||<\epsilon$

Corollary

Sequence $(x_{i}) \in \mathbb{R}^{n}$ converges to $x \in \mathbb{R}^{n} \iff$

$$\textrm{For } x_{i} = (x_{i}^{1},\dots,x_{i}^{n}) \textrm{ and } x = (x^{1},\dots,x^{n})$$ $$x_{i} \to x \iff \forall k \text{ } x_{i}^{k} \to x^{k} \text{ as } i \to \infty$$

1.2 Continuity

1.2.1 Open sets in Euclidean Spaces

Definition - Open Ball

Open ball of radius $r$ is

\[B_{r}(x) = \{y \in \mathbb{R}^{n} : ||x-y||<r\}\]

Definition 1.2 - Open Sets

A set $U \subseteq \mathbb{R}^{n}$ is called open if

$\forall x \in U, \exists r > 0$ such that $B_{r}(x) \subseteq U$

1.2.2 Continuity at a point/on an open set

Definition 1.3 - Continuity at a point

Let $A\subset\mathbb{R}^{n}$ an open set, with $f: A \to \mathbb{R}^{n}$ $f$ continuous at $p \in A$ if

\[\textcolor{RoyalBlue}{\forall \epsilon > 0, \exists \delta > 0 \text{ s.t } ||x-p|| < \delta \implies ||f(x)-f(p)|| < \epsilon}\]

$f$ is (pointwise) continuous on $A\subseteq \mathbb{R}^{n} \iff$ continuous $\forall p \in A$, we write $f$ is continuous.

For small enough $\delta$, we have $f(B_{\delta}(p)) \subseteq B_{\epsilon}(f(p))$

Theorem 1.2 - Composition of continuous functions
Let $A \subseteq \mathbb{R}^{n}$ open, $B \subseteq \mathbb{R}^{m}$ open and suppose $f: A \to B$ continuous at $p\in A$, and $g: B \to \mathbb{R}^{l}$ continuous at $f(p)$

$ \implies g \circ f: A \to \mathbb{R}^{l}$ continuous at p

Definition 1.4 - Limit of a function at a point

$A \subseteq \mathbb{R}^{n}$ an open set. $f$ a function
$f: A \to \mathbb{R}^{m}$, with $p \in A$ and $q \in \mathbb{R}^{m}$

Say $\lim_{x\to p}f(x) = q$ if

$\forall \epsilon > 0, \exists \delta > 0 \text{ s.t } \forall x \in A \text{ with } 0 < \lvert\lvert x-p\rvert\rvert <\delta \text{ we have } \lvert\lvert f(x) - p\rvert\rvert < \epsilon$

$f$ continuous at $p \iff \lim_{x\to p}f(x) = q$

Theorem 1.3 - Algebra of Limits
Suppose $A \subseteq \mathbb{R}^{n}$ open, with $p \in A$ and $f,g: A \to \mathbb{R}^{n}$

$\lim_{x\to p}f(x) = F $ and $\lim_{x\to p}g(x) = G$

Then:

$\lim_{x\to p}(f(x) + g(x)) = F + G$
$\lim_{x\to p}(f(x)g(x)) = FG$
If $G\neq 0$ then $\lim_{x\to p}\frac{f(x)}{g(x)} = \frac{F}{G}$

1.3 Derivative of a map of Euclidean Spaces

1.3.1 Derivative of a linear map

Lemma 1.5
The map $f:(a,b) \to \mathbb{R}$ differentiable at $p \in (a,b) \iff \exists$ map of the form $A_{\lambda}(x) = \lambda(x-p) + f(p)$ for some $\lambda \in \mathbb{R}$ s.t

\[\lim_{x\to p}\frac{|f(x) - A_{\lambda}(x)|}{|x-p|} = 0\]

Notation

$h[v]$ for $h$ a linear map, $v$ a vector
$h(v)$, $h$ a map, $v$ a point in domain of $h$
$L(\mathbb{R}^{n};\mathbb{R}^{m})$ - Set of linear maps from $\mathbb{R}^{n} \to \mathbb{R}^{m}$

Definition 1.5 - Derivative in higher dimension

Suppose $\Omega \subset \mathbb{R}^{n}$ open. The map $f:\Omega \to \mathbb{R}^{m}$ differentiable at $p \in \Omega$ if $\exists$ a linear map $\Lambda \in L(\mathbb{R}^{n};R^{m})$ such that

$$\lim_{x\to p}\frac{||f(x) - (\Lambda[x-p] + f(p))}{||x-p||} = 0$$

We write

$$Df(p) := \Lambda$$

Calling $Df(p)$ the derivative of $f$ at $p$
$\Lambda$ a $m\times n$ matrix called the Jacobian

Lemma 1.6 - Differentiable then continuous
$\Omega \subset \mathbb{R}^{n}$ open
$f:\Omega \to \mathbb{R}^{m}$ differentiable at $p\in \Omega \implies f$ continuous at $p$

Theorem 1.7 - Uniqueness of Derivative
The derivative, if it exists, is unique

1.3.2 Chain Rule

Chain rule in $\mathbb{R}$
$f,g: \mathbb{R}\to \mathbb{R}, g$ differentiable at $p$, $f$ differentiable at $g(p)$.

Then $f \circ g$ differentiable at $p$ with

$$(f\circ g)'(p) = f'(g(p))g'(p)$$

Theorem 1.8 - Chain rule in higher dim.

$\Omega \subset \mathbb{R}^{n}$ open, $\Omega’ \subset \mathbb{R}^{m}$ open
With $g:\Omega \to \Omega’$ differentiable at $p \in \Omega$, $f:\Omega’ \to \mathbb{R}^{l}$ differentiable at $g(p) \in \Omega’$
Then $h = f \circ g: \Omega \to \mathbb{R}^{l}$, differentiable at $p$, s.t

\[Dh(p) = D(f(g(p))\circ Dg(p)\]

1.4 Directional Derivatives

1.4.1 Rates of change and Partial Derivatives

Definition - Directional Derivative

The directional derivative of $f$ at $p$ in the direction $v$ is

$$\frac{\partial f}{\partial v}(p) := \lim_{t\to 0}\frac{1}{t}[f(p+vt)-f(p)] = Df(p)[v]$$

Definition - Partial derivatives

We can find any directional derivative at $p$, given we know the partial derivatives of $f$

\[D_{i}f(p) = \frac{\partial f}{\partial e_{i}}(p)\]

In $\mathbb{R}^{3}$ we have,

\[Df(p)[v] = \begin{pmatrix} \ D_{1}f(p)\quad D_{2}f(p)\quad D_{3}f(p)\quad \end{pmatrix} \begin{pmatrix} v^{1}\\ v^{2} \\ v^{3} \end{pmatrix}\]

Definition - Gradient

Gradient of $f$ at $p$

$$\nabla f(p) = \begin{pmatrix} D_{1}f(p)\\ D_{2}f(p) \\ D_{3}f(p) \end{pmatrix}\qquad Df(p) = (\nabla f(p))^{t}$$

Theorem 1.9 - Jacobian
Suppose $\Omega \subset \mathbb{R}^{n}$ open and $f: \Omega \to \mathbb{R}^{m}$ of the form

$$f(x) = \begin{pmatrix} f^{1}(x),f^{2}(x),\dots,f^{m}(x) \end{pmatrix}$$

If $f$ differentiable for some $p \in \Omega$ Then Jacobian of $f$ at $p$ is:

$$Df(p) = \begin{pmatrix} D_{1}f^{1}(p) & \dots & D_{n}f^{1}(p)\\ \vdots & \ddots & \vdots\\ D_{1}f^{m}(p) & \dots & D_{n}f^{m}(p) \end{pmatrix}$$

1.4.2 Relation between partial derivatives and differentiability

Theorem 1.12
Let $\Omega \subset \mathbb{R}^{n}$ open, $f:\Omega\to \mathbb{R}$. Suppose the partial derivatives:

$$D_{i}f(x) := \lim_{t\to 0}\frac{f(x+te_{i}-f(x)}{t}$$

exist $\forall x \in \Omega$, with each map $x \mapsto D_{i}f(x)$ continuous at $p, \forall i$$\implies$ $f$ is differentiable at $p$

1.5 Higher Derivatives

1.5.1 Higher derivatives as linear maps

Can think of the differential of $f$, $Df(p)$ as a map

$$Df: \Omega \to L(R^{n};R^{m}) = \Omega \to \mathbb{R}^{mn}$$ $$\quad p \mapsto Df(p)$$

if map $Df$ is continuous $\implies f:\Omega \to \mathbb{R}$ is continuously differentiable

Definition - Higher derivative

If $Df: \Omega \to \mathbb{R}^{mn}$ differentiable at $p$, denote derivative of $Df$ as $DDf(p): \mathbb{R}^{n} \to \mathbb{R}^{nm}$

$$DDf(p) \in L(\mathbb{R}^{n};\mathbb{R}^{nm}) = L(\mathbb{R}^{n};L(\mathbb{R}^{n};\mathbb{R}^{m}))$$

Where $DDf(p)$ is a linear map$\mathcal{L} \in L(\mathbb{R}^{n};L(\mathbb{R}^{n};\mathbb{R}^{m}))$, satisfying:

$$\lim_{x\to p}\frac{||Df(x) - Df(p) - \mathcal{L}[x-p]||}{||x-p||} = 0$$

$DDf(p)$ takes an $n$-vector to a $m\times n$ matrix

Definition - Continuously differentiable

$f:\Omega \to \mathbb{R}^{m}$ is $k$-times dfferentiable with all continuous derivatives $\implies$ $f$ is $k$-times continuously differentiable

Testing for $k$-times differentiability
For $f = \begin{pmatrix} f^{1}(x),f^{2}(x),\dots,f^{m}(x) \end{pmatrix}$
If $f$ differentiable at $p\in \Omega$ $\implies$ we have partial derivatives $D_{i}f^{j}: \Omega \to \mathbb{R}$.
If $Df$ differentiable, then $2^{\text{nd}}$ partial derivatives exist

$$D_{k}D_{i}f^{j}(p) := \lim_{t\to 0}\frac{D_{i}f^{j}(p+te_{k})-D_{i}f^{j}(p)}{t}$$

Easy to check these exist and are continuous $\implies$ $k$-times differentiability at $p$

1.5.2 Symmetry of mixed partial derivatives

Theorem 1.13 - Schwartz’ Theorem
Suppose $\Omega \subset \mathbb{R}^{n}$ open and $f:\Omega \to \mathbb{R}$ differentiable $\forall p \in \Omega$
Suppose also, for $i,j \in {1,\dots,n}, 2^{\text{nd}}$ partial derivatives $D_{i}D_{j}f$ and $D_{j}D_{i}f$ exist and are continuous $\forall p \in \Omega$

$$\forall p \in \Omega, D_{i}D_{j}f(p) = D_{j}D_{i}f(p)$$

Definition - Hessian

The matrix of $2^{\text{nd}}$ partial derivatives at the point $p$

$$\text{Hess } f(p) = [D_{i}D_{j}f(p)]_{i,j =1,\dots,n}$$

Schwartz’ Theorem says Hess $f(p)$ is a symmetric matrix

1.5.3 Taylor’s Theorem

Definition - Multi-inidices

Multi-index $\alpha \in (\mathbb{N})^{n}, \alpha = \begin{pmatrix} \alpha_{1},\dots,\alpha_{n} \end{pmatrix}$

We define $|a| = \sum_{i=1}^{n}\alpha_{i}$ and

$$D^{\alpha}f := (D_{1})^{\alpha_{1}}(D_{2})^{\alpha_{2}}\dots(D_{n})^{\alpha_{n}}f,$$

And for a vector $h = \begin{pmatrix} h_{1},\dots,h_{n} \end{pmatrix}$

$$h^{\alpha} := (h^{1})^{\alpha_{1}}(h^{2})^{\alpha_{2}}\dots(h^{n})^{\alpha_{n}}$$

Also

$$\alpha ! := \alpha_{1}!\alpha_{2}!\dots\alpha_{n}!$$

helpful examples

$D^{(0,3,0)}f(p) = D_{2}^{3}f(p)$
$D^{(1,0,1)}f(p) = D_{1}D_{3}f(p)$
$(x,y,z)^{(2,1,5)} = x^{2}y^{1}z^{5}$

Theorem 1.14 - Taylor’s Theorem in higher dim.
Suppose $p \in \mathbb{R}^{n}$ and $f: B_{r}(p) \to \mathbb{R}$ a $k$-times continuously differentiable $\forall q \in B_{r}(p)$, for some $k \geq 1 \in \mathbb{N}$

Then $\forall h \in \mathbb{R}^{n}$ with $||h|| < r$ We have

$$f(p+h) = \sum_{|\alpha| \leq k-1}\frac{h^{\alpha}}{\alpha!}D^{\alpha}f(p) + R_{k}(p,h)$$

Sum over all $\alpha = \begin{pmatrix} \alpha_{1},\dots,\alpha_{n} \end{pmatrix}$
with $|\alpha| \leq k-1$ and remainder term

$$R_{k}(p,h) = \sum_{|\alpha| = k}\frac{h^{\alpha}}{\alpha!}D^{a}f(x)$$

for some $x$ s.t $0 < ||x-p||< ||h||$
Evidently

$$\lim_{h\to 0}\frac{|R_{k}(p,h)|}{||h||^{k-1}} = 0$$

1.6 Inverse & Implicit Function Theorem

1.6.1 Inverse Function Theorem

Theorem 1.15 - (Inverse Function Theorem)

Let $\Omega$ an open set in $\mathbb{R}^{n}$, $f: \Omega \to \mathbb{R}^{n}$ continuously differentiable on $\Omega$, $\exists q \in \Omega$ s.t $Df(q)$ invertible Then $\exists$ open sets $U \subset \Omega$ and $V \subset \mathbb{R}^{n}, q \in U, f(q) \in V$ s.t

$f:U \to V$, a bijection
$f^{-1}: V \to U$, continuously differentiable
$\forall y \in V$, $Df^{-1}(y) = [Df(f^{-1}(y))]^{-1}$

1.6.2 Implicit Function Theorem

Theorem 1.16 - (Implicit Function Theorem - Simple version)
$\Omega \subset \mathbb{R}^{2}$ open
$F: \Omega \to \mathbb{R}$ continuously differentiable and $\exists (x’,y’) \in \Omega$ s.t

$F(x’,y’) = 0$, and
$D_{2}F(x’,y’) \neq 0$

$\implies \exists$ open sets $A, B \subset \mathbb{R}$ with $x’ \in A, y’ \in B$ with a map $f:A \to B$ s.t

$(x,y) \in A \times B$ satisfies $F(x,y) = 0 \iff y = f(x)$ for some $x \in A$

with $f:A \to B$ continuously differentiable.

Definition - $C^{1}- $ diffeomorphism

$\Omega,\Omega’ \subset \mathbb{R}^{n}$ open.
Say $f:\Omega \to \Omega’$ a $C^{1}$-diffeormorphism, if $f:\Omega \to \Omega’$ a and $\forall x \in \Omega, Df(x)$ invertible
$\mathcal{D}$ the set of all $C^{1}-$diffeomorphisms from $\Omega \to \Omega$, a group under group law; composition.

1.6.4 Implicit Function Theorem - General Form

Theorem 1.17 - (Implicit Function Theorem)
$\Omega \subset \mathbb{R}^{n}, \Omega’ \subset \mathbb{R}^{m}$ open sets
$F: \Omega \times \Omega’ \to \mathbb{R}^{m}$ continuously differentiable on $\Omega \times \Omega’$ and sps $\exists (a,b) \in \Omega \times \Omega’$ s.t

$f(p) = 0$ and,
$m \times n$ matrix

$$(D_{n+j}f^{i}(p)), \qquad 1 \leq i, j\leq m$$

invertible

$\implies \exists$ open sets $A \subset \Omega, B \subset \Omega’$ with $a \in A, b \in B$ with a map $g:A \to B$ s.t

$g(x,y) = 0 $ for some $(x,y) \in A \times B \iff y = g(x)$ for some $x \in A$

with $g:A \to B$ continuously differentiable.

2 Metric and Topological Spaces

2.1 Metric Spaces

2.1.1 Motivation + Definition

Definition 2.1 - Metric

$X$ an arbitrary set
Metric a function $d: X \times X \to \mathbb{R}$ satisfying:

$\forall x,y \in X;\ d(x,y) \geq 0, d(x,y) = 0 \iff x = y$ (positivity)
$\forall x,y \in X;\ d(x,y) = d(y,x)$ (symmetry)
$\forall x,y,z \in X d(x,y) \leq d(x,z) + d(z,y)$ (triangle inequality)

Definition 2.2 - Metric space

Pair of a set and metric; $M = (X,d)$
Call elements of $X$ points,with $d(x,y)$ distance between $x,y$ w.r.t $d$

Definition

$C([a,b]) = \{f:[a,b] \to \mathbb{R}| f:[a,b] \to \mathbb{R}$ continuous $\}$

2.1.2 Examples of metrics

$d_{2}(x,y) = \lvert\lvert x-y\rvert\rvert$; Euclidean metric on $\mathbb{R}^{n}$
$d_{\text{disc}}(x,y) = \begin{cases} 0, & x=y\ 1, & x \not = y \end{cases}$
$d_{\infty}(x,y) = \sup_{k\geq 1}\lvert x^{k} - y^{k}\rvert$
$d_{\infty}(f,g) = \text{max}_{a\leq t\leq b}\lvert f(t) - g(t)\rvert $ where $f,g \in C([a,b])$ (supremum/uniform metric)

Definition 2.3. Induced metrics

$(X,d)$ a metric space
$Y \subseteq X$, define $ d\rvert_{Y}: Y \times Y \to \mathbb{R}$ as $d|_{Y}(x,y) = d(x,y)\ \forall x,y \in Y$

Definition 2.3. Metric Subspace

Say $(Y,d\rvert_{Y})$ a metric subspace of $(X,d)$

Definition 2.4. Product metric space

$(X_{1},d_{1})$ and $(X_{2},d_{2})$ metric spaces.
define metric using $d_{1},d_{2}$ $d: (X_{1} \times X_{2}) \times (X_{1} \times X_{2}) \to \mathbb{R}$.
$(X_{1} \times X_{2}, d)$ a product metric space.

2.1.3 Normed Vector Spaces

Definition 2.5. Norm in Metric Spaces

$V$ a vector space on $\mathbb{R}$. Say $||\cdot||: V \to \mathbb{R}$ a [norm] on $V$ if

$\forall v \in V,\ \lvert\lvert v\rvert\rvert \geq 0$ and $\lvert\lvert v\rvert\rvert = 0 \iff v = 0$
$\forall v \in V, \forall \lambda \in \mathbb{R},\ \lvert\lvert\lambda v\rvert\rvert = \lvert\lambda\rvert\cdot\lvert\lvert v\rvert\rvert$
$\forall u,v \in V,\ \lvert\lvert u+v\rvert\rvert \leq \lvert\lvert u\rvert\rvert + \lvert\lvert v\rvert\rvert$

Definition - Normed Vector Space

A pair of a vector space $(V, ||\cdot||)$
note $||\cdot||$ is a metric on $V \implies$ normed vector space a metric space.

2.1.4 Open sets in metric spaces

Definition 2.6. Open ball in metric spaces

$(X,d)$, with $x \in X, \epsilon \in \mathbb{R}; \epsilon > 0$\

Ball radius $ \epsilon;\ B_{\epsilon}(x) = \{ x' \in X | d(x,x') < \epsilon\}$

Notation

$B_{\epsilon}(x,X,d)$

Definition 2.7. Open set in metric space

$(X,d)$ a metric space. $U \subseteq X$ open in $(X,d)$ if:

\[\forall u \in U,\ \exists \delta > 0 \in \mathbb{R}\text{ s.t } B_{\delta}(u) \subset U\]

Definition 2.8. Topologically equivalent

$d_{1},d_{2}$ metrics on a set $X$ topologically equivalent if: $\forall\ U \subseteq X,\ U \text{ open in } (X,d_{1}) \iff U \text{ open in } (X,d_{2})$

2.1.5 Convergence in Metric Spaces

Definition 2.9. Convergence in Metric Spaces

$(X,d)$ a metric space. $(x_{n})_{n\geq 1}$ a sequence in $X$.

Say $(x_{n}) \to x \in (X,d)$ if

$$\forall\ \epsilon > 0, \exists N \in \mathbb{N}\text{ s.t } \forall\ n \geq N, d(x,x_{n})< \epsilon$$

Lemma 2.7

if $(x_n)$ converges in $(X,d)$ $\implies$ limit is unique

Corollary - $d_{1},d_{2}$ topologically equivalent $\iff (x_n)$ converges in $(X,d_1)$ and $(X,d_2)$

2.1.6 Closed sets in metric spaces

Definition 9. Closed set in Metric Spaces

$(X,d)$ a metric space. $V \subseteq X$ a set.
$V$ closed in $(X,d)$ if $\forall\ (x_n) \in V$ s.t $(x_n) \to x$ convergent in $(X,d) \implies x \in V$

Theorem 2.9.

$(X,d)$ a metric space. $V \subseteq X$

$V$ closed in $(X,d) \iff X\backslash V$ open in $(X,d)$

Lemma 2.10

Intersection of closed sets in $(X,d)$ is a closed set in $(X,d)$
Finite union of closed sets in $(X,d)$ a closed set in $(X,d)$

Interior, isolated, limit, and boundary points in metric spaces

Definition 2.11. - 2.12.

$(X,d)$ a metric space, $V \subset X,\ x \in X$

$x$ an interior/inner point of $V$ if $$\exists \delta > 0,\ \text{ s.t } B_{\delta}(x) \subset V$$
1. Interior of $V$; $V^{\circ}$ - ${v \in V : v \text{ an interior point of } V}$
$x$ a limit/accumulation point of $V$ if $$\forall \delta > 0, (B_{\delta}(x) \cap V)\backslash\{x\} \neq \emptyset$$
Note: not all limit points of $V$ are in $V$
1. Closure of $V$; $\bar{V}$ - $V \cup {v \text{ a limit point of } V}$
$x$ a boundary point of $V$ if $$\forall \delta > 0, B_{\delta} \cap V \neq \emptyset \text{ and } B_{\delta}(x)\backslash V \neq \emptyset$$
1. Boundary of $V$; $\partial V$ - ${v \in X : v \text{ a boundary point of } V}$
$x$ an isolated point of $V$ if $$\exists \delta > 0, \text{ s.t } V \cap B_{\delta}(x) = \{x\}$$

Lemma 2.11

$(X,d)$ a metric space, $V \subseteq X$
$x \in X$ a limit point of $V \iff \exists$ sequence in $V \backslash { x }$ converging to $x$.

Definition 2.13. Dense and Seperable subsets

$(X,d)$ a metric space

$V \subseteq X$ dense in $X$ if $\bar{V} = X$
$(X,d)$ seperable if, $\exists$ dense countable subset of $X$

2.1.8 Continuous maps of metric spaces

Definition 2.14. Continuity in metric spaces

$(X,d_{X}), (Y,d_{Y})$ metric spaces.
$f: X \to Y$ a map

$f$ continuous at $x \in X$ if $\forall \epsilon > 0, \exists \delta > 0 \text{ s.t } \forall x' \in X \text{ s.t } d_{X}(x',x) < \delta, d_{Y}(f(x),f(x')) < \epsilon$
$f: X \to Y$ continuous if $f$ continuous $\forall x \in X$
$f: X \to Y$ uniformly continuous if $f$ continuous $\forall x \in X$ with $\delta = \delta(\epsilon)$ not depending on $x$

Theorem 2.12.

$(A_{1},d_1),(A_2,d_2)$ metric spaces
$f: A_1 \to A_2$ continuous $\iff$ pre-image of any open set in $A_2$ is an open set in $A_1$
$f: A_1 \to A_2$ continuous $\iff$ pre-image of any closed set in $A_2$ is a closed set in $A_1$

Theorem 2.13.

$(X,d_X), (Y,d_Y)$ metric spaces
$f: X \to Y$ a map; $f \text{ continuous at } x \in X \iff \text{ for any sequence } (x_n) \to x;\ f(x_n) \to f(x) \text{ in } (Y,d_Y)$

Definition 2.15. Homeomorphism

$(X_1,d_1),(X_2,d_2)$ metric spaces.

$f: X_1 \to X_2$ a homeomorphism if
- $f: X_1 \to X_2$ a bijection
- $f: X_1 \to X_2$ and $f^{-1}:X_2 \to X_1$ continuous
Say $(X_1,d_1),(X_2,d_2)$ homeomorphic if $\exists$ homeomorphism from $X_1$ to $X_2$

Definition 2.16.

$(X,d_X),(Y,d_Y)$ metric spaces with $f:X\to Y$\

$f$ is Lipschitz if $\exists$ constant $M > 0$ s.t $\forall x_1,x_2 \in X, d_Y(f(x_1),f(x_2)) \leq M \cdot d_X(x_1,x_2)$
$f$ is bi-Lipschitz if $\exists$ constants $M_1,M_2 > 0$ s.t $\forall x_1,x_2 \in X$ $$M_2 \cdot d_X(x_1,x_2) \leq d_Y(f(x_1),f(x_2)) \leq M_1 \cdot d_X(x_1,x_2)$$
Corollary; any bi-Lipschitz map is injective
$f$ an isometry/distance preserving if $\forall x_1,x_2 \in X;$ $$d_Y(f(x_1),f(x_2)) = d_X(x_1,x_2)$$

2.2 Topological Spaces

2.2.2 Topology on a set

Definition 2.15.. Topology

$A$ an arbitrary set. $\tau$ a collection of subsets of $A$
$\tau$ a topology on $A$ if:

$\emptyset \in \tau$ and $A \in \tau$
$G_{\alpha} \in \tau$ for $\alpha$ in a (finite) set $I$ $\implies \bigcup_{\alpha \in I}G_{\alpha} \in \tau$
$G_{1},G_{2},\dots,G_{m} \in \tau\implies \bigcap_{i=1}^{m}G_{i} \in \tau$

A topological space; $(A,\tau)$ a pair of a set $A$ and topology $\tau$ on $A$. Each element in $\tau$ an open set in $(A,\tau)$
$U$ a neighbourhood of $a$ if $U \in \tau$ and $a \in U$

Example 2.25.. Some Topologies

Coarse topology - $A$ arbitrary set, $\tau= {\emptyset, A}$
Induced topology - $(X,d)$ a metric space, with $\tau$ the collection of all open sets in $(X,d)$
Order Topology - $A = \mathbb{R}$ with $\tau$ collection of subsets of $\mathbb{R}$ of form $(a,+\infty),\ a \in \mathbb{R}\cup { -\infty,+\infty }, (+\infty,+\infty) := \emptyset$
Discrete Topology - $A$ arbitrary, $\tau= \mathcal{P}(A)$
Product topology -

Definition. Metrisable topological space

Say topological space $(X,\tau)$ metrisable if $\exists$ metric on $X$ which induces a topology $\tau$.

Definition. Induced and Subspace topology

$(X,\tau)$ a topological space. $Y \subset X$

\[\tau_{Y} = \{U \cap Y | U \in \tau\}\]

$\tau_{Y}$ the induced topology on $Y$ from $(X,\tau)$
$(Y,\tau_{Y})$ has the subspace topology induced from $(X,\tau)$

Definition 2.18. Stronger topology

$A$ a set, with $\tau_1,\tau_2$
Say $\tau_1$ stronger (or finer) than $\tau_2$ if $\tau_2 \subset \tau_1$

Lemma 2.14.
$(A,\tau)$
A set $G \subset A$ open $\iff \forall\ x \in G,\ \exists$ neighbourhood of $x$ contained in $G$

Definition 2.19. Interior in Topological space

$(A,\tau)$ a topological space. $\Omega \subseteq A$
$z \in \Omega$ an interior point of $\Omega$ if

\[\exists U \in \tau\text{ s.t } z \in U \text{ and } U \subset \Omega\]

Interior of $\Omega; \Omega^{\circ}$ = $ { z \in \Omega \lvert z$ an interior point of $ \Omega }$

Properties of interior

$S \subset T \implies S^{\circ} \subset T^{\circ}$
$S$ open in $A \iff S = S^{\circ}$
$S^{\circ}$ largest open set contained in $S$

2.2.3 Convergence, and Hausdorff property

Definition 2.20. Convergence in Topological Spaces

$(A,\tau)$ a topological space. $(x_n)_{n\geq1}$ a sequence in $A$
$(x_n)$ converges in $(A,\tau)$ if

\[\exists x \in A \text{ s.t } \forall\ G \in \tau\text{ with } x \in G,\ \exists N \in \mathbb{N}, \text{ s.t } \forall n \geq N, x_n \in G\]

Definition 2.21. Hausdorff

$(A,\tau)$ called Hausdorff if: $\forall x,y \in A\ x \neq y,\ \exists \text{ open set } U,V \text{ s.t } x \in U, y \in V \text{ and } U \cap V = \emptyset$ Say $U$ and $V$ seperate $x$ and $y$

Theorem 2.14.

$(A,\tau)$ a Hausdorff topological space. $(x_n)$ a sequence in $A$.
if $(x_n)$ convergent in $(A,\tau) \implies$ limit is unique.

2.2.4 Closed sets in topological spaces

Definition 2.22. Closed set in Topological space

$(A,\tau)$ a topological space.
$V \subseteq A$. Say $V$ closed in $(A,\tau) \iff A\backslash V \in \tau$

Lemma 2.17.
$(A,\tau)$ a topological space $\implies \emptyset$ and $A$ closed in $(A,\tau)$

intersection of closed sets in $(A,\tau)$ is a closed set in $(A,\tau)$
union of a finite number of closed sets in $(A,\tau)$ is a closed set in $(A,\tau)$

Definition 2.23. Limit/Accumulation point in Topological Spaces

$(A,\tau),$ a topological space, $S\subseteq A$
$x \in A$ a limit/accumulation point of $S$ if

\[\forall\ U \text{ a neighbourhood of } x,\ (S \cap U)\backslash\{x\} \neq \emptyset\]

$x$ not necessarily in $S$
Closure of $S, \bar{S}$$= S \cup { x \in A | x \text{ a limit point of } S}$

Lemma
$S$ closed in $(A,\tau) \iff S = \bar{S}$

2.2.5 Continuous maps on topological spaces

Definition 2.24. Continuity in topological space

$(X,\tau_X),(Y,\tau_Y)$ with $f: X \to Y$
$f$ continuous on $X$ if:

$\forall$ open sets

\[U \in Y,\ f^{-1}(U) \text{ open in } X\]

Theorem 2.20.

$(X,\tau_X),(Y,\tau_Y)$ with $f: X \to Y$
$f$ continuous $\iff$ pre-image of closed set in $Y$ is closed in $X$

Theorem 2.21.

$(X,\tau_X),(Y,\tau_Y),(Z,\tau_Z)$
$f: X \to Y, g:Y\to Z$ continuous $\implies g \circ f : X \to Z$ continuous

Definition 2.25. Homeomorphisms in Topological space

$f: X \to Y$ a homeomorphism is $f: X \to Y$ bijective with $f$ and $f^{-1}$ continuous

Definition 2.25. Topologically equivalent in Topological space

$(X,\tau_X),(Y,\tau_Y)$ topologically equivalent/homeomorphic if $\exists$ homeomorphism from $X \to Y$

2.3 Connectedness

2.3.1 Connected sets

Definition 2.26. Disconnected sets

For $(X,d)$ a metric space, consider $T \subseteq X$. $T$ disconnected ,if $\exists$ open sets $U,V \in X$ s.t:

$U \cap V = \emptyset$
$T \subseteq U \cup V$
$T \cap U \neq \emptyset$ and $T \cap V \neq \emptyset$

Set connected if not disconnected. i.e for any 2 of the properties that hold from above the 3rd cannot.

Lemma 2.23.
$(X,d)$ a metric space. $T \subseteq X$

$T$ disconnected $\iff\ \exists$ continuous $f:T \to \mathbb{R}\text{ s.t } f(T) = \{0,1\}$

Theorem 2.22.

Consider $(\mathbb{R},d)$, $S \subseteq \mathbb{R}$

$S$ connected $\iff S$ an interval

2.3.2 Continuous maps + Connected sets

Theorem 2.27.

$(A,d_{1})$ and $(A,d_{2})$ metric spaces. $f: A_1 \to A_2$ continuous map $S \subset A$ connected $\implies f(S)$ connected

Corollary 2.28 $f:(X,d_X) \to (Y,d_Y)$ a homeomorphism

$X$ connected $\iff Y$ connected

Theorem 2.29.

$(X,d)$ connected metric space, $f: X \to \mathbb{R}$ continuous. Assume $\exists a,b \in X$ s.t $f(a) <0, f(b) > 0 \implies \exists c \in X$ s.t $f(c) = 0$

2.3.3 Path Connected Sets

Definition 2.28. Path

Under $(X,d)$ given $a,b \in X$
Path from $a \to b$ a continuous map $f: [0,1] \to X$ s.t $f(0) = a, f(1) = b$

Definition 2.29. Path Connected

$(X,d)$ path connected if $\forall a,b \in X, \exists$ path from $a\to b$ in $X$

Theorem 2.30.

if $(X,d)$ path connected $\implies$ connected

2.4 Compactness

2.4.1 Compactness by covers

Definition 2.30. Covers

$(X,d)$ a metric space. $Y \subseteq X$

collection $R$ of open subsets of $X$ an open cover for $Y$ if $$Y \subseteq \bigcup_{v \in R}v$$
Given open cover $R$ for $Y$
Say $C$ a sub-cover of $R$ for $Y$ if $C \subseteq R$ and $Y \subseteq \bigcup_{v \in R}v$
Open cover $R$ for $Y$ is a finite cover if $R$ has finitely many elements.

Definition 2.31. Compact

$(X,d)$ a metric space
$Y \subseteq X$ compact in $(X,d)$ if every open cover for $Y$ has a finite sub-cover.

Proposition 2.32.
$a,b \in \mathbb{R},\ a \leq b$ in $(R,d_1)$ we have $[a,b]$ compact

Proposition 2.33.
$(X,d)$ a metric space, $Y \subseteq X$
$X$ compact, $Y$ closed $\implies Y$ compact.

Theorem 2.34.

$(X,d)$ a metric space $Y \subset X$

$Y$ compact $\implies Y $ closed

Theorem 2.35.

$(X,d_X),(Y,d_Y)$ metric spaces. Considering $(X\times Y,d)$
$d((x_{1},y_{1}),(x_{2},y_{2})) = d_{1}(x_1,x_2) + d_2(y_1,y_2)$
$X, Y$ compact $\implies (X \times Y,d)$ compact

Corollary.
$[a_1,b_1]\times[a_2,b_2]\dots\times[a_{n-1},b_{n-1}]\times[a_n,b_n]$ compact in $\mathbb{R}^{n}$

Definition 2.32. Bounded

$(X,d)$ non-empty metric space, $Z \subseteq X$
$Z$ bounded in $(X,d)$ if $\exists M \in \mathbb{R}$ s.t $\forall x,y \in Z; d(x,y) \leq M$
$S$ arbitrary set. $f: S \to X$ bounded if $f(S)$ bounded in $X$

Lemma 2.37.
$(X,d)$ compact metric space $\implies X$ bounded

Theorem 2.36. Heine-Borel

Consider $(\mathbb{R}^{n},d_{2})$, $X \subseteq \mathbb{R}^{n}$
$X$ compact $\iff X$ closed and bounded

2.4.2 Sequential Compactness

Definition 31. Sequentially compact

$(X,d)$ sequentially compact, if for every sequence in $X$ has convergent subsequence in $(X,d)$

\[\forall (x_n)_{n\geq 1} \in X,\ \exists (x_{n_k})_{k\geq 1},\ x \in X \text{ s.t } x_{n_k} \to x\]

Lemma 2.39.

$(X,d)$ a metric space. with sequence $\ (x_{n}){n\geq 1} \text{ s.t } \exists (x{n_k}){k\geq 1},\ x \in X \text{ s.t } x{n_k} \to x $.

\[\iff \exists x \in X \text{ s.t } \forall \epsilon > 0 \text{ there are infinitely many } i \text{ s.t } x_{i} \in B_{\epsilon}(x)\]

Theorem 2.41. Bolzanno-Weierstrass

Any bounded sequence in $\mathbb{R}^{n}$ has convergent subsequence.

Theorem 2.40 + 2.42.

$(X,d)$ metric space.

$X$ Compact $\iff X$ Sequentially Compact

2.4.3 Continuous maps + Compact Sets

Theorem 2.41.

$(X,d_X),(Y,d_Y)$ metric spaces.
$f: X\to Y$ a continuous map if $Z \text{ compact in } X \implies f(Z) \text{ compact in }Y$

Corollary 2.44.
$(X,d_X),(Y,d_Y)$ metric spaces, $f:X \to Y$ a homeomorphism

\[\implies X \text{ compact } \iff Y \text{ compact }\]

Theorem 2.45.

Every continuous map from compact metric space to a metric space is uniformly continuous.

Corollary 2.46. $f:[a,b] \to \mathbb{R}$ continuous $\implies$ $f$ uniformly continuous

Theorem 2.47.

$(X,d_X)$ compact, $f:X \to \mathbb{R}$ continuous $\implies f$ bounded above and below attaining its upper & lower bounds

Theorem 2.48.

$f:\mathbb{R}\to \mathbb{R}$ continuous w.r.t Euclidean metrics on domain and range.
$\forall\ [a,b]$ we have $f([a,b])$ of the form $[m,M]$ for $m,M \in \mathbb{R}$

2.5 Completeness

2.5.1 Complete metric spaces & Banach space

Definition 32. Cauchy Sequence

$(X,d)$ a metric $(x_{n})_{n\geq 1}$ sequence in $X$

Say $(x_{n})_{n\geq 1}$ a Cauchy sequence in $(X,d)$ if

\[\forall \epsilon > 0, \exists N_{\epsilon} \in \mathbb{N}\text{ s.t } \forall n,m \geq N_{\epsilon} d(x_{n},x_{m}) < \epsilon\]

Definition 2.35. Complete & Banach

metric space $(X,d)$ complete if every Cauchy sequence in $X$ converges to a limit in $X$
Normed vector space $(V,\lvert\lvert\cdot\rvert\rvert)$ a Banach space if $V$ with induced metric space $d_{\lvert\lvert\cdot\rvert\rvert}$ a complete metric space.

Theorem 2.51.

Assume $(f_{n} : [a,b] \to \mathbb{R})_{n\geq 1}$ sequence of continuous functions converging uniformly to $f:[a,b] \to \mathbb{R}\implies f:[a,b] \to \mathbb{R}$ continuous

Theorem 2.52.

Metric space $(C([a,b]),d_{\infty})$ is complete or equivalently $(C([a,b]),\lvert\lvert\cdot\rvert\rvert_{\infty})$ a Banach space

Theorem 2.53.

$(X,d)$ a compact metric space $\implies (X,d)$ complete

2.5.2 Arzelà-Ascoli

Definition 2.36. Uniformly bounded & Uniformly equi-continuous

Let $\mathcal{C}$ a collection of functions $f:[a,b] \to \mathbb{R}$

Say collection $\mathcal{C}$ uniformly bounded if $\exists M$ s.t $\forall f \in \mathcal{C}$ and $\forall x \in [a,b] \implies |f(x)| < M$
Say collection $\mathcal{C}$ uniformly equi-continuous if $\forall \epsilon > 0, \exists \delta > 0$ s.t $\forall f \in \mathcal{C}$ and $\forall x_1,x_2 \in [a,b]$ s.t $|x_1-x_2| < \delta$ we have $|f(x_1) - f(x_2)| < \epsilon$

Theorem 2.54. Arzelà-Ascoli

Assume $\mathcal{C}$ collection of continuous functions $f:[a,b] \to \mathbb{R}$ if $\mathcal{C}$ uniformly bounded and uniformly equi-continuous $\implies$ every sequence in $\mathcal{C}$ has convergent subsequence in $(C([a,b],d_{\infty})$

2.5.3 Fixed point theorem

Definition 2.37. Contracting

$(X_{1},d_{1})$ and $(X_{2},d_{2})$, with $f: X_1 \to X_2$
Say $f$ contracting if $\exists K \in (0,1)$ s.t $\forall a,b \in X$ we have

\[d_{2}(f(a),f(b)) \leq K\cdot d_{1}(a,b)\]

Every contracting map is continuous.

Definition 2.37. Fixed point

$f:X\to X$ say $x \in X$ a fixed point of $f$ if $f(x) = x$

Theorem 2.55. Banach fixed point theorem

$(X,d)$ a non-empty complete metric space.
$f: X \to X$ a contracting map $\implies f$ has unique fixed point in $X$

Term II

Holomorphic Functions

Complex Numbers

Definition 1. $i$

\[i = \sqrt{-1},\quad i^2 = -1\]

Root of $x^{2} + 1 = 0$

Basic properties

\[z = x + iy, \quad Re(z) = x,\ Im(z) = y\]

The complex conjugate:

\[\bar{z} = x - iy\]

Polar Coordinates
$z = x+iy$

\[r = |z| = \sqrt{x^2 + y^2}\] \[x = r\cos\theta,\ y = r\sin\theta\] \[z = r(\cos\theta + i\sin\theta\]

De-Moivre’s Formula

\[z^{n} = r^{n}(\cos(n\theta) + i\sin(n\theta)),\ n \in \mathbb{Z}^{+}\]

Eulers’s Formula $e^{i\theta} = (\cos\theta + i\sin\theta)$

Sets in Complex Plane

Definition 2. Discs in $\mathbb{C}$

Open Disc : $D_{r}(z_{0})$ $= { z \in \mathbb{C}: z-z_{0} < r}$
Boundary of Disc : $C_{r}(z_{0})$ $= { z \in \mathbb{C}: z-z_{0} = r}$
Unit Disc : $\mathbb{D}$ $= { z \in \mathbb{C}: z < 1}$

Definition 3. Interior Point

$\Omega \in \mathbb{C}, z_{0}$ an interior point of $\Omega$ if $\exists r > 0$ s.t $D_{r}(z_0) \subset \Omega$

Definition 4.

Set $\Omega$ open if $\forall \omega \in \Omega$, $\omega$ an interior point

Definition 5.

Set $\Omega$ closed if $\Omega^{C} = \mathbb{C}\backslash\Omega$ open
Closed $\iff$ contains all its limit points.

Definition 6. Closure

Closure of $\Omega$ = $\bar{\Omega} = { \Omega \cup \text{ limit points of } \Omega}$

Definition 7. Boundary

Boundary of $\Omega = \underbrace{\bar{\Omega}}{\text{Closure}}\backslash\underbrace{\partial\Omega}{\text{interior}}$

Definition 8. Bounded

$\Omega$ bounded if $\exists M > 0$ s.t $|\omega| < M\ \forall \omega\in \Omega$

Definition 9. Diameter

\[diam(\Omega) = \sup_{z,w \in \Omega}|z-w|\]

Definition 10. Compact

$\Omega$ compact if closed and bounded

Theorem 1.

\[\begin{aligned} \Omega \text{ compact} & \iff \text{every sequence} \{z_{n}\} \subset \Omega \text{ has a subsequence convergent in } \Omega \\ & \iff \text{every open covering of } \Omega \text{ has a finite subcover}\end{aligned}\]

Theorem 2.

if $\Omega_1 \supset \Omega_2 \supset \dots \Omega_n \supset \dots$ a sequence of non-empty compact sets
s.t $\lim_{n\to \infty}diam(\Omega_n) \to 0$ $\implies \exists ! w \in \mathbb{C},\ w \in \Omega_n\ \forall n$

Definition 11. Connected

Open set $\Omega$ connected $\iff$ any 2 points in $\Omega$ joined by curve $\gamma$ entirely contained in $\Omega$

Complex Functions

Definition 12.

$\Omega_1, \Omega_2 \subset \mathbb{C}$ $f: \Omega_1 \to \Omega_2$ a mapping $\Omega_1 \to \Omega_2$ if

\[\forall z = x + iy \in \Omega_1\]

$\exists ! w = u + iv \in \Omega_2,\ s.t\ w = f(z)$ We have $w = f(z) = u(x,y) + iv(x,y)$
$u,v : \mathbb{R}^{2} \to \mathbb{R}$

Definition 13.

$f$ defined on $\Omega_1 \subset \mathbb{C}$ $f$ continuous at $z_0 \in \Omega$ if $\forall \epsilon > 0 \exists \delta > 0\ s.t\ |z-z_{0}| < \delta \implies |f(z) - f(z_{0})| < \epsilon$ $f$ continuous if continuous $\forall z \in \Omega$

Complex Derivative

Definition 14. Holomorphic

$\Omega_1, \Omega_2 \subset \mathbb{C}$ open sets
$f: \Omega_1 \to \Omega_2$
Say $f$ differentiable/holomorphic at $z_{0}$ if

\[\lim_{h\to 0}\frac{f(z_{0} + h) - f(z_0)}{h} = f'(z_0) \text{ exists}\]

$f$ holomorphic on open set $\Omega$ if holomorphic at every point of $\Omega$

Lemma
$f$ holomorphic at $z_0 \in \Omega \iff \exists\ a \in \mathbb{C}$ s.t

\[f(z_0 + h) - f(z_0) - ah = h\Psi(h)\]

For $\Psi$ a function defined for all small $h$ with $\lim_{h\to 0}\Psi(h) = 0$, $a = f’(z_0)$

Corollary
$f$ holomorphic $\implies$ $f$ continuous

Proposition
$f,g$ holomorphic in $\Omega$ $\implies$

$(f+g)’ = f’ + g’$
$(fg)’ = f’g + fg’$
$g(z_0) \neq 0 \implies (\frac{f}{g})’ = \frac{f’g - fg’}{g^2}$
$f:\Omega \to V,\ g:V \to \mathbb{C}$ holomorphic
$\implies [g \circ f(z)]’ = g’(f(z))f’(z)\ \forall z \in \Omega$

Cauchy-Riemann equations

tcolorbox $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ $u'_x = v'_y \qquad u'_y = -v'_x$

Definition 15.

\[\frac{\partial}{\partial z} = \frac{1}{2}\left( \frac{\partial}{\partial x} + \frac{1}{i}\frac{\partial}{\partial y}\right)\qquad \frac{\partial}{\partial \bar{z}} = \frac{1}{2}\left( \frac{\partial}{\partial x} - \frac{1}{i}\frac{\partial}{\partial y}\right)\]

Theorem 3.

$f(z) = u(x,y) + iv(x,y)\quad z = x+iy$
$f$ holomorphic at $z_0$ $\implies$

\[\frac{\partial f}{\partial \bar{z}}(z_0) = 0\qquad f'(z_0) = \frac{\partial f}{\partial z}(z_0) = 2\frac{\partial u}{\partial z}(z_0)\]

Theorem 4.

$f = u+ iv$ complex-valued function on open set $\Omega$
$u,v$ continuously differentiable, satisfying Cauchy-Riemann equations $\implies f$ holomorphic on $\Omega$ with $f’(z) = \frac{\partial f}{\partial z}(z)$

Cauchy-Riemann equations in polar

For $f = u + iv$ we have

\[u'_r = \frac{1}{r}v'_\theta \qquad v'_r = -\frac{1}{r}u'_\theta\]

Power Series

Definition 16. Power Series

Of the form

\[\sum_{n = 0}^{\infty}a_n z^{n} \quad a_n \in \mathbb{C}\]

Series converge at $z$ if $S_N(z) = \sum_{n=0}^{N}a_n z^{n}$ has limit $S(z) = \lim_{n\to \infty}S_{N}(z)$

Theorem 5.

Given power series $\sum_{n=0}^{\infty}a_{n}z^{n}$, $\exists 0 \leq R \leq \infty$ s.t

if $\lvert z\rvert < R \implies$ series converges absolutely
$\lvert z\rvert > R \implies$ series diverges

\[\frac{1}{R} = \limsup_{n\to \infty}|a_n|^{1/n}\qquad \text{(Radius of Convergence)}\]

Theorem 6.

\[f(z) = \sum_{n = 0}^{\infty}a_n z^{n}\]

Defines holomorphic function on its disc of convergence. With,

\[f'(z) = \sum_{n = 1}^{\infty}na_nz^{n-1}\]

with same radius of convergence as $f$.

Power series infinitely differentiable in the disc of convergence, achieved through term-wise differentiation.

Definition 17. Entire

A function said to be entire if holomorphic $\forall z \in \mathbb{C}$

Elementary functions

Exponential function

\[e^z = e^x\cos y + ie^x \sin y \qquad z = x + iy \in \mathbb{C}\]

Properties

$y = 0 \implies e^z = e^x$
$e^z$ is entire
$g(z)$ holomorphic
$\implies \frac{\partial}{\partial z}e^{g(z)} = e^{g(z)}g’(z)$
$z_1,z_2 \in \mathbb{C}\quad e^{z_1 + z_2} = e^{z_1}e^{z_2}$
$\lvert e^z\rvert = \lvert e^x\rvert \lvert e^{iy} = e^x\sqrt{\cos^{2}x + \sin^{2}(x)} = e^x$
$(e^{iy})^{n} = e^{iny}$
$arg(z) = \arctan(y/x)$
$arg(e^z) = y + 2\pi k,\quad k \in \mathbb{Z}$

Trigonometric functions

Definition 18. - Trig functions

\[\cos z = \frac{1}{2}\left( e^{iz} + e^{-iz}\right) \qquad \sin z = \frac{1}{2i}\left( e^{iz} - e^{-iz}\right)\]

Properties

$\sin z, \cos z$ are entire
$\frac{\partial}{\partial z}\sin z = \cos z \quad \frac{\partial}{\partial z}\cos z = -\sin z$
$\sin^{2}z + \cos^{2}z = 1$
$\sin(z_{1} \pm z_{2}) = \sin z_1 \cos z_2 \pm \cos z_1 \sin z_2$
$\cos(z_1 \pm z_2) = \cos z_1 \cos z_2 \mp \sin z_1 \sin z_2$

Logarithmic Functions

Definition 19. - Log Functions

\[log(z) = ln|z| + i\arg(z) = log(r) + i(\theta + 2\pi k) \quad z \neq 0,\ k \in \mathbb{Z}\]

$\log(z)$ a multi-valued function

Definition 20.

$\mathop{\mathrm{Log}}(z) = \ln|z| + i\mathop{\mathrm{Arg}}(z)\ $ for $\mathop{\mathrm{Arg}}(z)$ principal value $\in [-\pi,\pi]$

Properties

$\log(z_1 z_2) = \log(z_1) + \log(z_2)$
$\mathop{\mathrm{Log}}(z)$ holomorphic in $\mathbb{C}\backslash {(\infty,0]}$

Powers

Definition 21.

$\alpha \in \mathbb{C}$ define $z^{\alpha} := e^{\alpha\log(z)}$ as a multi-valued function

Definition 22.

Principal value of $z^{\alpha},\ \alpha \in \mathbb{C}$ as $z^{\alpha} = e^{\alpha\mathop{\mathrm{Log}}(z)}$ Properties

$z^{a_{1}}z^{a_{2}} = z^{a_{1} + a_{2}}$

Cauchy’s Integral Formula

Parametrised Curve

Definition 23.

Parametrised curve a function $z(t): [a,b] \to \mathbb{C}$

Smooth if $z’(t)$ exists and is continuous on $[a,b]$ with $z’(t) \neq 0 \forall t \in [a,b]$

Taking one-sided limits for $z’(a), z’(b)$.

Piecewise-smooth if $z$ continuous on $[a,b]$ and if $\exists$ finitely many points $a = a_{0} < a_1 < \dots < a_n = b$ s.t $z(t)$ smooth on $[a_{k},a_{k+1}]$

\[z:[a,b] \to \mathbb{C}\quad \tilde{z}:[c,d] \to \mathbb{C}\]

equivalent if $\exists$ continuously differentiable bijection $s \to t(s)$ from $[c,d]$ to $[a,b]$ s.t $t’(s) > 0$ and $\tilde{z}(s) = z(t(s))=$

Definition 24. Path integral

Path integral given smooth $\gamma \subset \mathbb{C}$ parametrised by $z:[a,b] \to \mathbb{C}$.
$f$ continuous function on $\gamma$

\[\int_{\gamma}f(z)dz = \int_{a}^{b}f(z(t))z'(t)dt\]

independent of choice of parametrization.
If $\gamma$ piece-wise smooth $\int_{\gamma}f(z)dz = \sum_{k=0}^{n-1}\int_{a_{k}}^{a_{k+1}} f(z(t))z'(t)dt$

Definition 25.

Define curve $\gamma^{-}$ obtained by reversing orientation of $\gamma$
Can take $z^{-}:[a,b]\to \mathbb{C}$ s.t $z^{-}(t) = z(b + a -t)$

Definition 26. Closed Curve

Smooth/piece-wise smooth curve closed if $z(a) = z(b)$ for any parametrisation.

Definition 27. Simple Curve

Smooth/piece-wise smooth curve simple if not self-intersecting

\[z(t) \neq z(s) \text{ unless } s = t \in [a,b]\]

Integration along Curves

Definition 28. Length of smooth curve

\[\text{Length}(\gamma) = \int_{a}^{b}|z'(t)|dt = \int_{a}^{b}\sqrt{x'(t)^{2} + y'(t)^{2}}dt\]

Theorem 7. Properties of Integration

$\int_{\gamma}af(z) + bg(z) dz = a\int_{\gamma}f(z)dz + b\int_{\gamma}g(z)dz$
$\gamma^{-}$ reverse orientation of $\gamma$ $\implies \int_{\gamma}f(z)dz = -\int_{\gamma^-}f(z)dz$
M-L inequality $\left|\int_{\gamma}f(z)dz\right| \leq \sup_{z\in\gamma}|f(z)|\cdot\mathop{\mathrm{length}}(\gamma) = \int_{a}^{b}\sqrt{x'(t)^{2} + y'(t)^{2}}dt$

Primitive Functions

Definition 29. Primitive

A Primitive for $f$ on $\Omega \subset \mathbb{C}$ a function $F$ holomorphic on $\Omega$ s.t $F’(z) = f(z)\ \forall z \in \Omega$

Theorem 8.

Continuous function $f$ with primitive $F$ in open set $\Omega$ and curve $\gamma$ in $\Omega$ from $w_1 \to w_2$

\[\int_{\gamma}f(z)dz = F(w_2) - F(w_1)\]

Corollary

$\gamma$ closed curve in open set $\Omega$ $f$ continuous and has primitive in $\Omega \implies$

\[\int_{\gamma}f(z)dz = 0\]

Corollary
$\Omega$ with $f’ = 0 \implies f$ constant

Properties of Holomorphic functions

Theorem 9.

Let $\Omega \subset \mathbb{C}$ open set
$T \subset \Omega$ a triangle whose interior contained in $\Omega$

\[\implies \int_{T}f(z)dz = 0\]

for $f$ holomorphic in $\Omega$

Corollary
$f$ holomorphic on open set $\Omega$ containing rectangle $R$ in its interior $\implies \int_{R}f(z)dz = 0$

Local existence of primitives and Cauchy-Goursat theorem in a disc

Theorem 10.

Holomorphic functions in open disc have a primitive in that disc

Corollary - (Cauchy-Goursat Theorem for a disc)

$f$ holomorphic in disc $\implies \int_{\gamma}f(z)dz = 0$ for any closed curve $\gamma$ in that disc

Corollary

Suppose $f$ holomorphic in open set containing circle $C$ and its interior

\[\implies \int_{C}f(z)dz = 0\]

Homotopies and simply connected domains

Definition 30. Homotopic

$\gamma_{0},\gamma_{1}$ homotopic in $\Omega$ if $\forall s \in [0,1], \exists$ curve $\gamma \subset \Omega$ with $\gamma_{s}(t)$ s.t

\[\gamma_{s}(a) = \alpha \qquad \gamma_{s}(b) = \beta\]

$\forall t \in [a,b]: \gamma_{s}(t)\vert_{s = 0} = \gamma_{0}(t) \quad \gamma_{s}(t)\vert_{s = 1} = \gamma_{1}(t)$

With $\gamma_{s}(t)$ jointly continuous in $s \in [0,1]$ and $t \in [a,b]$

Theorem 11.

$\gamma_{0},\gamma_{1}$ homotopic, $f$ holomorphic

\[\int_{\gamma_{0}}f(z)dz = \int_{\gamma_{1}}f(z)dz\]

Definition 31.

Open set $\Omega \subset \mathbb{C}$ simply connected if any 2 pair of curves in $\Omega$ with shared end-points homotopic.

Theorem 12.

Any holomorphic function in simply connected domain has a primitive.

Corollary - (Cauchy-Goursat Theorem)

$f$ holomorphic in simply connected open set $\Omega$

\[\implies \int_{\gamma}f(z)dz = 0\]

for any closed piecewise-smooth curve $\gamma \subset \Omega$

Theorem 13. (Deformation Theorem)

$\gamma_{1}$ and $\gamma_2$, 2 simple closed piecewise-smooth curves with $\gamma_2$ lying wholly inside $\gamma_1$

$f$ holomorphic in domain containing region between $\gamma_1,\gamma_2$

\[\implies \int_{\gamma_1}f(z)dz = \int_{\gamma_2}f(z)dz\]

Cauchy’s Integral Formulae

Theorem 14. (Cauchy’s Integral Formula)

$f$ holomorphic inside and on simple closed piecewise-smooth curve $\gamma$

$\forall z_0 \text{ interior } to \gamma$

\[f(z_0) = \frac{1}{2\pi i}\int_{\gamma}\frac{f(z)}{z-z_0}dz\]

Theorem 15. (Generalised cauchy’s integral formula)

$f$ holomorphic in open set $\Omega$.

$\gamma$ simple,closed piecewise-smooth $\subset \Omega$

$\forall z$ interior to $\gamma$

\[\implies \frac{d^n f(z)}{dz^n} = \frac{n!}{2\pi i}\int_{\gamma}\frac{f(t)}{(t-z)^{n+1}}dt\]

Corollary
$f$ holomorphic $\implies$ all its derivatives are too.

Applications of Cauchy’s integral formula

Corollary - (Liouville’s theorem)
if an entire function bounded $\implies$ $f$ constant

Theorem 16. (Fundamental theorem of algebra)

Every polynomial of degree $> 0$ with complex coefficients has at least one zero.

Corollary Every polynomial $P(z) = a_n z^n + \dots + a_0$ of degree $n \geq 1$ has precisely $n$ roots in $C$

Theorem 17. (Morera’s theorem)

Suppose $f$ continuous in open disc $D$ s.t $\forall$ triangle $T \subset D$ $\int_{T}f(z)dz = 0 \implies f \text{ holomorphic }$

Taylor + Maclaurin Series

Theorem 18. (Taylor’s expansion theorem

$f$ holomorphic in $\Omega$, $z_0 \in \Omega$

\[\implies f(z) = f(z_0) = f'(z_0)(z-z_0) + \frac{f''(z_0)}{2!}(z-z_0)^{2} + \dots\]

Valid in all circles ${ z: \lvert z - z_0 \rvert < r} \subset \Omega$

Definition 32. (Taylor Series)

\[f(z) = f(z_0) + f'(z_0)(z-z_0) + \frac{f''(z_0)}{2!}(z-z_0)^{2} + \dots = \sum_{i=0}^{\infty}\frac{f^{(n)}(z_0)}{n!}(z-z_0)^{n}\]

Definition 33. (Maclaurin Series)

Taylor series for $z_0 = 0$

\[f(z) = \sum_{n=0}^{\infty}\frac{f^{n}(0)}{n!}z^n\]

Sequences of holomorphic functions

Theorem 19.

if ${f_n}_{n = 1}^{\infty}$ a sequence of holomorphic functions converging uniformly to $f$ in every compact subset of $\Omega \implies f$ holomorphic in $\Omega$

Corollary

\[F(z) = \sum_{n=1}^{\infty}f_{n}(z)\]

$f_n$ holomorphic in $\Omega \subset \mathbb{C}$

Given series converges uniformmly in compact subsets of $\Omega$ $\implies F(z)$ holomorphic

Theorem 20.

Sequence ${f_{n}}_{n=1}^{\infty} \xrightarrow[unif]{} f$ in every compact subset of $\Omega$

$\implies$ sequence ${f’n}{n=1}^{\infty} \xrightarrow[unif]{} f’$ in every compact subset of $\Omega$

Holomorphic functions defined in terms of integrals

Theorem 21.

Let $F(z,s)$ defined for $(z,s) \in \Omega \times [0,1]$
$\Omega \subset \mathbb{C}$ open set. Given $F$ satisfies

$F(z,s)$ holomorphic in $\Omega \forall s$
$F$ continuous on $\Omega\times[0,1]$

$\implies f(z) := \int_{0}^{1}F(z,s)ds$ holomorphic

Schwarz reflection principle

Definition 34.

$\Omega \subset \mathbb{C}$ open and symmetric w.r.t real line $z \in \Omega \iff \bar{z} \in \Omega$

Definition 35.

\[\textcolor{green}{\Omega^{+}} = \{ z\in \Omega: Im(z) > 0\} \quad \textcolor{green}{\Omega^{-}} = \{ z\in \Omega: Im(z) < 0\} \quad \textcolor{green}{I} = \{ z\in \Omega: Im(z) = 0\}\]

Theorem 22. (Symmetry Principle)

$f^+,f^-$ holomorphic in $\Omega^+,\Omega^-$ respectively.
Extend continuously to $I$ s.t $f^+(x) = f^-(x) \quad \forall x \in I$

\[f(z) := \begin{cases} f^+(z), & z \in \Omega^+\\ f^+(z) = f^-(z), & z \in \Omega^{-}\\ f^-(z), & z \in \Omega^- \end{cases} \quad \text{holomorphic}\]

Theorem 23. (Schwarz relfection principle)

$f$ holomorphic in $\Omega^{+}$ extend continuously to $I$ s.t $f$ real-valued on $I$
$\implies \exists F$ holomorphic in $\Omega$ s.t $F\rvert_{\Omega^{+}} = f$

Meromorphic Functions

Complex Logarithm

Theorem 24.

$\Omega$ simply connected, $1\in \Omega, 0 \not\in \Omega$
$\implies$ in $\Omega$ there is a branch of logarithm

\[F(z) = \log_{\Omega}(z)\]

Satisfying

$F$ holomorphic in $\Omega$
$e^{F(z)} =z \ \forall z \in \Omega$
$F(r) = \log(r),\ r\in \mathbb{R}$ close to $1$

Theorem 25.

Holomorphic $f$ has $0$ of order $m$ at $z_0$
$\iff$ can be written in form $f(z) = (z-z_0)^m g(z)$ $g$ holomorphic at $z_0$, $g(z_0) \neq 0$

Corollary
$0$ s of non-constant holomorphic function are isolated.
Every zero has neighbourhood, inside of which it is the only $0$

Laurent Series

Definition 36.

Laurent Series for $f$ at $z_0$, where series converge $\begin{align*} f(z) &= \sum_{-\infty}^{\infty} a_{n}(z-z_{0})^{n}\\ &= \dots + a_{-2}(z-z_{0})^{-2} + a_{-1}(z-z_{0})^{-1} + a_{0} + a_{1}(z-z_{0})^{1} + a_{2}(z-z_{0})^{2} + \dots \end{align*}$

Theorem 26. (Laurent Expansion theorem)

$f$ holomorphic in anunulus $D = { z: r < |z-z_{0}| < R }$
$\implies f(z)$ expressed in form $f(z) = \sum_{-\infty}^{\infty} a_{n}(z-z_{0})^{n}$

\[a_{n} = \frac{1}{2\pi i}\int_{\gamma}\frac{f(\eta)}{(\eta - z_{0})^{n+1}} d\eta\]

$\gamma$ simple,closed piecewise smooth curve in $D$ with $z_0$ in its interior.

Poles of holomorphic functions

Definition 37.

$z_0$ a singularity of complex function $f$
if $f$ not holomorphic at $z_0$, but every neighbourhood of $z_0$ has at least 1 holomorphic point.

Definition 38.

Singulartiy $z_0$ is isolated if $\exists$ neighbourhood of $z_0$, where it is the only singularity.

Definition 39.

$f$ holomorphic with isolated singularity $z_0$
Considering Laurent expansion valid in some annulus

\[f(z) = \sum_{-\infty}^{\infty} a_{n}(z-z_{0})^{n}\]

$\implies$

$a_n = 0 \ \forall n < 0 \implies z_0$ a [removable singularity]
$a_n = 0 \forall n < -m, m \in \mathbb{Z}^{+}, a_{-m} \neq 0 \implies z_0$ pole of order $m$
$a_{n} \neq 0$ for infinitely many negative $n$ $\implies$ $z_0$ a [essential singularity]

Theorem 27.

$f$ has pole of order $m$ at $z_0$ $\iff$ written in form

\[f(z) = \frac{g(z)}{(z - z_0)^m}\]

$g$ holomorphic at $z_0$, $g(z_0) \neq 0$

Residue Theory

Definition 40.

Let $f(z) = \sum_{-\infty}^{\infty} a_{n}(z-z_{0})^{n}$ for $0 < |z-z_0| < R$ the Laurent series for $f$ at $z_0$
Residue of $f$ at $z_0$ is

\[\implies Res[f,z_0] = a_{-1}\]

Theorem 28.

$\gamma \subset { z : 0< |z-z_0| <R}$ simple closed piecewise-smooth curve containing $z_0$ $\implies Res[f,z_0] = \frac{1}{2\pi i}\int_{\gamma}f(z) dz$

Lemma

$f$ holomorphic inside and on
$\gamma$ a simple closed piecewise-smooth curve with $z_i$ the singularities in its interior.

\[\int_{\gamma}f(z)dz = 2\pi i \cdot \sum_{z_0 \text{ a singularity }} \text{Res}[f,z_0]\] \[\text{Res}[f,z_0] = \lim_{z\to z_0}\frac{d^{m-1}}{dz^{m-1}}\frac{f(z)(z-z_0)^{m}}{(m-1)!} \quad m = ord(z_0)\]

Theorem 29.

$f$ holomorphic function inside and on simple closed piecewise-smooth curve $\gamma$ except at the singularities
$z_1,\dots,z_n$ in its interior

\[\implies \int_{\gamma}f(z) dz = 2\pi i \sum_{j=1}^{n} Res[f,z_j]\]

The argument principle

Theorem 30. (Princple of argument)

$f$ holomorphic in open $\Omega$, except for finitely many poles.
$\gamma$ simple closed piecewise-smooth curve in $\Omega$ not passing through poles or zeroes of $f$

\[\implies \int_{\gamma} \frac{f'(z)}{f(z)}dz = 2\pi i(N-P)\]

$N = \sum order(zeroes)$ $P = \sum order(poles)$ ————————– ————————-

Theorem 31. (Rouche’s Theorem)

$f,g$ holomorphic in open $\Omega$
$\gamma \subset \Omega$ simple closed piecewise-smooth curve with interior containing only points of $\Omega$
if $|g(z)| < |f(z)|, z\in \gamma$

\[\implies \sum_{0s \text{ of } f+g \text{ in } \gamma} order(zeros) = \sum_{0s \text{ of } f \text{ in } \gamma} order(zeros)\]

Definition 41.

Mapping open if maps open sets $\mapsto$ open sets

Theorem 32. (Open mapping theorem)

if $f$ holomorphic and non-constant in open $\Omega \subset \mathbb{C}$ $\implies f \text{ open }$

Remark
$f$ open $\implies |f|$ open

Theorem 33. (Max modulus principle)

$f$ non-constant holomorphic in open $\Omega \subset \mathbb{C}$
$\implies f$ cannot attain maximum in $\Omega$

Corollary
$\Omega$ open with closure $\bar{\Omega}$ compact
$f$ holomorphic on $\Omega$ and continuous on $\bar{\Omega}$ $\sup_{z\in \Omega}|f(z)| \leq \sup_{z\in \bar{Omega}\backslash \Omega}|f(z)|$

Evaluation of definite integrals

This subsection is to be filled in.

Harmonic Functions

Harmonic functions

Definition 42.

$\varphi = \varphi(x,y): \mathbb{R}^2 \to \mathbb{R},\ x,y \in \mathbb{R}$
$\varphi$ harmonic in open $\Omega \subset \mathbb{R}^2$ if

\[\begin{aligned} \underbrace{\Delta\varphi(x,y)}_{\text{laplace operator}} &:= \frac{\partial^2 \varphi}{\partial x^2}(x,y) + \frac{\partial^2 \varphi}{\partial y^2}(x,y)\\ &:= \varphi''_{xx}(x,y) + \varphi''_{yy}(x,y)\\ &:= 0 \end{aligned}\]

Theorem 34.

$f(z) = u(x,y) + iv(x,y)$ holomorphic in open $\Omega\subset \mathbb{C}$
$\implies u,v$ harmonic

Theorem 35. (Harmonic conjugate)

$u$ harmonic in open disc $D \subset \mathbb{C}$
$\implies \exists$ harmonic $v$ s.t $f = u+ iv$ holomorphic in $D$
$v$ the harmonic conjugate to $u$

Remark
In simply connected domain $\Omega \subset \mathbb{R}^2$ every harmonic function $u$ has harmonic conjugate $v$ s.t

\[v(x,y) = \int_{\gamma} \left( -\frac{\partial u}{\partial y}dx + \frac{\partial u}{\partial x}dy \right)\]

Integral independent of path, by Green’s theorem as $u$ harmonic and $\Omega$ simply connected.

Properties of real + imaginary parts of holomorphic function

Theorem 36.

Assume $f = u + iv$ holomorphic on open connected $\Omega \subset \mathbb{C}$

\[\begin{aligned} u(x,y) &= C\\ v(x,y) &= K\\ C,K &\in \mathbb{R} \end{aligned}\]

If $(1)$ and $(2)$ have same solution $(x_0,y_0)$ and $f’(x_0 + iy_0) \neq 0$
$\implies$ curve defined by $(1)$ orthogonal to curve defined by $(2)$

Preservation of angles

Definition 43.

Consider smooth curve $\gamma \subset \mathbb{C}$
$z(t) = x(t) + iy(t) \quad t \in [a,b]$
$\forall t_0 \in [a,b]$ we have direction vector

\[\begin{aligned} L_{t_0} &= \{ z(t_0) + tz'(t_0): t \in \mathbb{R}\}\\ &= \left\{ x(t_0) = tx'( t_0 + i(y(t_0) + ty'(t_0)) : t \in \mathbb{R}\right\} \end{aligned}\]

For $\gamma_1, \gamma_2$ curves parameterised by functions $z_1(t),z_2(t)$, $t\in[0,1]$ s.t $z_1 (0) = z_2 (0)$
Define angle between $\gamma_1, \gamma_2$ as angle between tangents

\[\arg z_{2}'(0) - \arg z_{1}' (0)\]

Theorem 37. (Angle preservation theorem)

$f$ holomorphic in open $\Omega \subset \mathbb{C}$
Given $\gamma_1,\gamma_2$ inside $\Omega$, parameterised by $z_1 (t), z_2 (t)$
Take $z_0 = z_1 (0) = z_2 (0)$ with $z_1 ‘(0), z_2 ‘(0), f’(z_0) \neq 0$

\[\underbrace{\arg z_{2}'(t) - \arg z_{1}'(t)}_{\text{angle between } z_{1}(0),z_{2}(0)} \rvert_{t = 0} = \underbrace{\arg f(z_{2}'(t)) - \arg f(z_{1}'(t))}_{\text{angle between }f(z_{1}(0)),f(z_{2}(0))} \rvert_{t = 0} (\mod 2\pi)\]

Definition 44.

$\Omega$ open $\subset \mathbb{C}$
$f:\Omega \to \mathbb{C}$ conformal if holomorphic in $\Omega$ and if $f’(z) \neq 0 \forall z \in \Omega$
Conformal mappings preserve angles.

Definition 45.

Holomorphic function a local injection on open $\Omega \subset \mathbb{C}$
if $\forall z_0 \in \Omega, \exists D = \{ z: |z-z_0| < r \} \subset \Omega \text{ s.t } f: D\to f(D) \text{ an injection}$

Theorem 38.

$f:\Omega \to \mathbb{C}$ local injection and holomorphic

\[\implies f'(z) \neq 0 \quad \forall z \in \Omega\]

Inverse of $f$ defined on its range holomorphic
$\implies$ inverse of conformal mapping also holomorphic

Möbius Transformations

Definition 46.

Mobius Transformation / Bilinear transformation a map

\[f(z) = \frac{az + b}{cz + d} \quad a,b,c,d \in \mathbb{C}, ad -bc \neq 0\]

Remark
Mobius Transformations holomorphic except for simple pole $z = -\frac{d}{c}$ with derivative

\[f'(z) = \frac{ad-bc}{(cz+b)^2}\]

$\implies$ mapping conformal for $\mathbb{C}\backslash {-\frac{d}{c}}$

Theorem 39.

Inverse of mobius transformation a mobius transformation
Composition of mobius transformations a mobius transformations

Corresponding to matrix multiplication and inverses

Definition 47. (Special/Simple mobius tranformations)

(M1). $f(z) = az$ Scaling and rotation by $a$
(M2).. $f(z) = z+ b$ Translation by $b$
(M3).. $f(z) = \frac{1}{z}$ Inverse and reflection w.r.t real axis

Theorem 40.

Every mobius transformations a composition of (M1),(M2),(M3).

Corollary
Mobius transformations:

circles $\mapsto$ circles
interior points $\mapsto$ interior points

Straight lines, considered to be circles of infinite radius

Cross-ratios Mobius Transformations

Theorem 41.

$w = f(z)$ a Mobius Transformation
s.t distinct $(z_1,z_2,z_3) \mapsto (w_1,w_2,w_3)$

\[\implies \left( \frac{z-z_1}{z-z_3} \right)\left( \frac{z_2-z_3}{z_2-z_1} \right) = \left( \frac{w-w_1}{w-z_3} \right)\left( \frac{w_2-w_3}{w_2-w_1} \right) \quad \forall z\]

Conformal mapping of half-plane to unit disc

Theorem 42.

\[\mathbb{D} = \{z: |z| <1\}\] \[\mathbb{H} = \{z = x +iy : Im(z) = y > 0\}\]

$w = f(z) = \frac{i-z}{i+z}$
$g(w) = \frac{1-w}{1+w}$

Riemann mapping theorem

Definition 48.

$\Omega \subset \mathbb{C}$ proper if non-empty and $\Omega\neq \mathbb{C}$

Theorem 43.

$\Omega$ proper and simply connected
if $z_0 \in \Omega \implies \exists !$ conformal $f: \Omega \to \mathbb{D}$ s.t $f(z_0) = 0$ and $f’(z_0) > 0$

Corollary
Any $2$ simply connected open subsets in $\mathbb{C}$ conformally equivalent.