MTH2003 Complex Analysis Pre-Lecture 2

• Recording.
• Lecture Notes.
• Lecture: MTH2003 Complex Analysis Lecture 2.
• Practical (MTH2003 Complex Analysis Practical 1).

Lots of definitions and notation - good for flashcards, but make sure they’re linked notes too (separate them). Could convert other linked definitions into flashcard definitions, too.

Open Sets, Closed Sets, and Their Boundary.

For the definitions of continuity, differentiability, and integrability, the concepts of open and closed intervals are required.

The analogous concepts for complex functions are open and closed subsets of the plane $\mathbb{C}$ .

The simplest subset of the plane is a disk, where a point $Z$ is inside a disk with radius $ϵ>0$ (using epsilon since it should be a “small” number) centred at $z_0$ if $|z-z_0|<{ϵ}^2$ (equivalent to $(x-x_0{)}^2-(y-y_0{)}^2<{ϵ}^2$). All $z$ points defined in this subset define the neighbourhood of $z_0\in \mathbb{C}$…

$$N_{ϵ}(z_0)=\left\{z\in \mathbb{C}:|z-z_0|<ϵ\right\}$$

A subset $D\in \mathbb{C}$ is called open if every point $z_0$ in $D$ has a neighbourhood inside $D$…

$$\forall z_0\in \mathbb{C}\exists ϵ>0:N_{ϵ}(z_0)\subset D$$

By definition, an open set doesn’t contain any “boundary” points (imagine a circle with no hard border - the same as normal intervals, but two-dimensional). Note that using disks, you can also call them open disks.

$$S=\left\{z\in \mathbb{C}:|z-a|<r\right\}$$

All points outside of an open set form a closed set, e.g. $\mathbb{C}\backslash N_{ϵ}(z_0)$.

The points inside of neighbourhoods, $z_0$, are called interior points.

$$\bar{S}=\left\{z\in \mathbb{C}:|z-a|\le r\right\}$$

By definition, each boundary point’s neighbourhood contains at least one point within and one point outside of the set.

The boundary notation is…

$$\delta S=\left\{z\in \mathbb{C}:|z-a|=r\right\}$$

The complement of a set is defined as the subset of everything outside of the set. This helps with the definition of a closed set: a subset is a closed set if its complement is an open set, $\bar{S}=S\cup \delta S$. This is a natural extension of the definition.

A region is a set consisting of an open set and an arbitrary number of its boundary points.

A region is bounded if it can be enclosed within a circle, closed if it contains all its boundary points, and compact if it’s both.

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EXAMPLES:

Let $S_1=\left\{z\in \mathbb{C}:|z-1|<\frac{1}{4}\right\}$…

• Interior points are defined as all points inside a circle with radius $\frac{1}{4}$ and centre $(1,0)=1+0i$.
• Boundary points are defined as $\delta S=\left\{z\in \mathbb{C}:|z-1|=\frac{1}{4}\right\}$.
• An open subset of $\mathbb{C}$ as any point $z_0$ in $S$ admits a neighbourhood $N_{ϵ}(z_0)$ that is entirely inside $S$.
• By extension, you can define a similar set with $\le$ which is closed, as well as show that the boundary $\delta S$ is closed.

DIAGRAM

Let $S_2=\left\{z\in \mathbb{C}:\text{Im}z\ge 1\right\}$…

DIAGRAM

Let $S_3=\left\{z\in \mathbb{C}:\text{Re}z>0\right\}$…

DIAGRAM

Let $S_4=\left\{z\in \mathbb{C}:y>x^2\right\}$…

DIAGRAM

Let $S_5=\left\{0\right\}$…

• Closed set, as the only existing neighbourhood is not entirely contained in $S_5$ by definition.

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Functions of a Complex Variable.

A function of a complex variables assigns a complex number $z$ a unique complex number $w$ such that $f(z)=w$, where $f$ is defined on an open subset of $\mathbb{C}$, $D$.

Examples: polynomial, trigonometric, exponential, rational, etc.

By definition, these functions can be expressed in real and imaginary parts:

$$f(z)=f(x+iy)=u(x,y)+iv(x,y)$$

where $u$ is the real part of the function and $v$ is the imaginary part.

Include a graph of these functions - two graphs, one with the domain and another with the image which the function translate between.

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EXAMPLES:

Polynomial…

$$\begin{array}{rl}f(z)=z^2=f(x+iy)& =(x+iy{)}^2\\ & =x^2+2ixy-y^2\\ & =(x^2-y^2)+i(2xy)\\ & \therefore u(x,y)=x^2-y^2,v(x,y)=2xy\end{array}$$

Rational…

$$\begin{array}{rl}g(z)& =\frac{1}{z+i}:D=\left\{z\in \mathbb{C}:z\ne -i\right\}=\mathbb{C}\backslash \left\{-i\right\}\\ & =\frac{1}{x+iy+i}\\ & =\frac{1}{x+i(y+1)}\\ & =\frac{x-i(y+1)}{[x+i(y+1)][x-i(y+1)]}\\ & =\frac{x}{x^2+(y+1{)}^2}+i\frac{-(y+1)}{x^2+(y+1{)}^2}\end{array}$$

Exponential…

$$\begin{array}{rl}{e}^z& =e^{x+iy}\\ & =e^x\cdot e^{iy}\\ & =e^x(\cos y+i\sin y)\\ & =e^x\cos y+i{e}^x\sin y\end{array}$$

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==Then describe the methods used, like using Euler’s formula which then also leads to the complex cos/sin function definitions with Euler’s number, as well as the hyperbolic functions and thus corresponding formulae - could even link this with Osborn’s Rule==.

Multi-valued Functions: The Logarithm.

Some of these complex “functions” are defined as inverses of other functions, like logarithms (inverse of $e^z$) and square roots (inverse of $z^2$). Hence, they’re not always single-valued - strictly, this invalidates them as functions, but their domain can be restricted to make them single-valued.

For example, the logarithm…

For any $z\ne 0$, the natural logarithm of $z$ is a complex number $w$ such that $e^w=z$. If $z=r{e}^{i\theta }$, then $\ln z$ (the usual logarithm of a positive real) is defined as…

$$w=\ln z=\ln (r{e}^{i\theta })=\ln r+i(\theta +2k\pi ):k=0,\pm 1,\pm 2,\dots$$

or

$$w=\ln z=\ln |z|+i\arg z$$

Clearly, this is multi-valued as $k$ can have an arbitrary number of values. This can hence be easily restricted to consider only one value of $k$ - often the principal value (or principal branch) where $k=0$. Hence,

$$\ln z=\ln r+i\theta :0\le \theta <2\pi$$

EXAMPLE: Finding all values of $\ln i$…

$$\begin{array}{rl}\ln i& =\ln |i|+i\arg (i)\\ & =0+i\left(\frac{\pi }{2}+2k\pi \right):\text{principal value }k=0\\ & =i\frac{\pi }{2}\end{array}$$