Algebra and Geometry in the Complex Plane
The Complex Plane
The complex number can be expressed as $z = x + yi$
- x is called the real part of z; $x = Re(z)$.
- y is called the imaginary part of z; $y = Im(z)$.
We also call the set of complex number as complex plane $\mathbb{C}$. Real number is the subset of the complex numbers.
Simple Algebra
Add
Suppose we have $z_1 = x + yi$ and $z_2 = u + vi$, we define the complex number adding as:
\[z_1 + z_2 = \underbrace{(x+u)}_{Re(z_1+z_2)} + \underbrace{(y+v)}_{Im(z_1+z_2)}i\]Modulus
Define the modulus of complext number as:
\[\vert z\vert = \sqrt{x^2 + y^2}\]Properties:
- $\vert z_1 z_2\vert = \vert z_1\vert\cdot\vert z_2\vert$
Inequalities:
- $-\vert z\vert \leq Re(z) \leq \vert z\vert$
- $-\vert z\vert \leq Im(z) \leq \vert z\vert$
- $\vert z+w\vert \leq \vert z\vert + \vert w\vert$
- $\vert z-w\vert \geq \vert z\vert - \vert w\vert$
$( z - w + w) \leq \vert z-w\vert + \vert w\vert $
Multiplication
Definition
\[z_1 * z_2 = (xu + yv) + (xv + yu)i\]Properties:
- associative: $(z_1z_2)z_3 = z_1(z_2z_3)$
- commutative: $z_1z_2= z_2z_1$
- distributive: $z_1(z_2 + z_3) =z_1z_2 + z_1z_3$
Conjugate
Definition
\[\bar{z} = x-yi\]Properties:
- $\bar{z}z = x^2 + y^2 = \vert z\vert ^2$
- $\bar{\bar{z}} = z$
- $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$
- $\vert \bar{z}\vert = \vert z\vert $
- $\frac{1}{z} = \frac{\bar{z}}{\bar{z}z} = \frac{\bar{z}}{\vert z\vert ^2}$
- $\frac{z}{w} = \frac{\bar{z}}{\bar{w}}$
Division
Definition
\[\frac{z}{w} = \frac{z\cdot \bar{w}}{w \cdot \bar{w}} = \frac{z\cdot \bar{w}}{\vert w\vert ^2}\]So that the numerator is real number.
Polar Representation and the Root of Complex Numbers
Polar Representation
Difinition
Similar to polar coordinates, the complex number can be expressed by radius $r$ and angle $\theta$ \(z = x+yi=r(\sin\theta + \cos\theta i)\)
where $r = \vert z\vert $ and $\tan \theta = \frac{y}{x}$.
The principal argument of z, called Arg z, is the value of $\theta \ \text{for which} -\pi \le \theta \leq \pi$.
Thus the $\theta$ is defined:
\[\theta = arg z = \{Arg z + 2\pi k: k = 0, \pm1, \pm2, ...\}\]We can also use simplified exponential notation:
\[z = r\exp(i\theta)\]Properties
Using exponential notation, we have the following properties:
- $\vert e^{i\theta}\vert = 1$
- $\overline{e^{i\theta}} = e^{-i\theta}$
- $e^{i\theta} \cdot e^{i\varphi} = e^{i(\theta + \varphi)}$
- $(e^{i\theta})^n = e^{in\theta}$
$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$
The multiplication in exponential form:
\[z \cdot w = r_1\cdot r_2 \cdot e^{i(\theta + \varphi)}\]Root of Complex Numbers
To find the n th root of $w = \rho e^{i\varphi}$, we can think reversively. Analytically, solve the following equation:
\[z^n = r^n e^{in\theta} = w\]We got
\[\begin{cases} r & = \sqrt[n]{\rho} \\ \theta &= \frac{\varphi + 2k\pi}{n}, \ k = 0,1,2 ... n-1 \end{cases}\]Topology in the Plane
Sets in the Complex Plane
Circles and Disk
We defined circles and disk centered at $z_0$ as:
\[\begin{aligned} B_r(z_0) &= \{ z \in \mathbb{C}: \text{z has distance less than }r\text{ from }z_0\}\\ &=\{ z \in \mathbb{C}: \vert z-z_0\vert < r\}\\ K_r(z_0) &= \{ z \in \mathbb{C}: \text{z has distance }r\text{ from }z_0\}\\ &=\{ z \in \mathbb{C}: \vert z-z_0\vert = r\} \end{aligned}\]Interior Points and Boundary Points
The points type is defined with respect to a set $E$.
A point $z_0$ is an interior point of E if:
\[\exists r, B_r(z_0) \subset E\]A point $z_0$ is a boundary point of E if:
\[\forall r, B_r(z_0) \text{ contains a point in E and a point not in E}\]We also define the set of all boundary points of E as $\partial E$
Open and Closed Sets
A set is open set if every one of its points is an interior point.
A set is closed set if it contains all of its boundary points.
Closure and Interior of a Set
Closure set of E is the set E toghther with all its boundary points:
\[\bar{E} = E \cup \partial E\]The interior set of E is $\dot{E}$ which contains all of interior points of E.
Connectedness
Intuitively, a set is connected if it is “in one piece”.
To make it more precise, we difine seperate set first:
$\text{Two sets X, Y in }\mathbb{C} \text{ are separated if there are disjoint open set U, V so that } X\subset U\text{ and }Y \subset V.$
Disjoint means that $U \cap V = \emptyset$
Then we define connected set as:
If it is impossible to find two separated non-empty sets whose union equals W, we call W as > connected.
It’s hard to check whether a set is connected!!! But for open set we have a simple rule to determine:
Let G be an open set in C. Then G is connected if and only if any two points in G can be joined in G by successive line segments.
Bounded Sets
We define bounded set as:
A set A in $\mathbb{C}$ is bounded if there exists a number R > 0 such that $A \subset B_R(0)$. If no such R exists then A is called unbounded.
