The Complex Plane

• Complex numbers: expressions of the form $z=x+iy.$ where
  • $x$ is called the real part of $z$; $\ x=\Re(z)$ or $(\operatorname{Re}z)$, and
  • $y$ is called the imaginary part of $z$; $\ y=\Im(z)$ or $(\operatorname{Im}z).$
• Set of complex numbers: $\mathbb{C}$ (the complex plane).
• Real numbers: subset of the complex numbers (those whose imaginary part is zero).
• The complex plane can be identified with $\mathbb{R}^2.$

Adding Complex Numbers

Thus
\[
\Re(z+w) = \Re(z) + \Re(w) \mbox{ and } \Im(z+w) = \Im(z) + \Im(w).
\]
Graphically, this corresponds to vector addition.

The Modulus of a Complex Number

Multiplication of Complex Numbers

• Motivation: $(x+iy)\cdot(u+iv)=(xu+ixv+iyu)+i^2yv$
• So we define:

• Example:
  \[(3+4i)(-1+7i)=(-3-28)+i(21-4)=-31+17i.\]
• The usual properties hold:
  • $(z_1z_2)z_3=z_1(z_2z_3)$ (associative)
  • $z_1z_2=z_2z_1$ (commutative)
  • $z_1(z_2+z_3)=z_1z_2+z_1z_3$ (distributive)

So What is i?

\[i=0+1i,\]

so

\[i^2=(0+1i)(0-1i)=(0\cdot0-1\cdot1)+i(0\cdot1+1\cdot0)=-1.\]

• $i^3=i^2\cdot i=-1\cdot i=-1$
• $i^4=i^2\cdot i^2=(-1)(-1)=1$
• $i^5=i^4\cdot i=i$
• $i^6=-1\ ...$

How Do You Divide Complex Numbers?

Suppose that $z=x+iy$ and $w=u+iv$. What is $\frac{z}{w}$ (for $w\neq0)?$

\begin{align*}
\frac{z}{w}&=\frac{x+iy}{u+iv}\\
&=\frac{(x+iy)(u-iv)}{(u+iv)(u-iv)}\\
&=\frac{(xu+yv)+i(-xv+yu)}{u^2+v^2+i(-uv+vu)}\\
&=\frac{xu+yv}{u^2+v^2}+i\frac{yu-xv}{u^2+v^2}.
\end{align*}

In particular:

\[\frac{1}{z}=\frac{1}{x+iy}=\frac{x-iy}{x^2+y^2}, \mbox{ as long as } z\neq 0.\]

The Complex Conjugate

Note the importance of the quantity $x-iy$ in the previous calculation!

Properties:

• $\bar{\bar{z}}=z$
• $\overline{z+w}=\bar{z}+\bar{w}$
• $|z|=|\bar{z}|$
• $z\bar{z}=(x+iy)(x-iy)=x^2+y^2=|z|^2$
• $\frac{1}{z}=\frac{\bar{z}}{z\bar{z}}=\frac{\bar{z}}{|z|^2}$

More Properties of the Complex Conjugate

• When is $z=\bar{z}?$
• $z+\bar{z}=(x+iy)+(x-iy)=2x$, so

\[
\Re(z)=\frac{z+\bar{z}}{2},\text{ similarly }\Im(z)=\frac{z-\bar{z}}{2i}.
\]

• $|z\cdot w| = |z|\cdot|w|$
• $\left(\frac{z}{w}\right)=\frac{\bar{z}}{\bar{w}}, (w\neq 0)$
• $|z|=0\text{ if and only if }z=0.$

Some Inequalities

• $-|z|\le\Re(z)\le|z|$
• $-|z|\le\Im(z)\le|z|$
• $|z+w|\le|z|+|w|\textit{ (triangle inequality)}$
• $|z-w|\ge|z|-|w|\textit{ (reverse triangle inequality)}$

The Fundamental Theorem of Algebra

Polar Coordinates

• Consider $z=x+iy\in\mathbb{C}, z\neq 0.$
• $z$ can also be described by the distance $r$ from the origin $(r=|z|)$ and the angle $\theta$ between the positive $x$-axis and the line segment from $0$ to $z.$
• $(r,\theta)$ are the polar coordinates of $z.$
• Relation between Cartesian and polar coordinates:
  • $x=r\ cos\ \theta$
  • $y=r\ sin\ \theta$
• ... so:\begin{align*}
  z &=x+iy\\
  &=r\ cos\ \theta\ + i\ r\ sin\ \theta\\
  &=r(cos\ \theta\ + i\ sin\ \theta).\\
  \end{align*}
  This is called the Polar representation of $z$.

The Argument of a Complex Number

• $z=x+iy=r(cos\ \theta+i\ sin\ \theta).$
• $r=|z|$ is easy to find, but how to find $\theta$? Note: $\theta$ is not unique!

• arg $z=\{\text{Arg }z+2\pi k: k=0,\pm 1,\pm 2,...\},z \neq 0.$
• Examples:
  • Arg $i=\frac{\pi}{2},$
  • Arg $1=0,$
  • Arg$(-1)=\pi,$
  • Arg$(1-i)=-\frac{\pi}{4},$
  • Arg$(-i)=-\frac{\pi}{2},...$

Exponential Notation

• Convenient notation: $e^{i\theta}=cos\ \theta+i\ sin\ \theta.$
• So $z= r(cos\ \theta + i\ sin\ \theta)$ becomes $z=re^{i\theta}$, the polar form of $z.$
• Note: $e^{i(\theta+2\pi)}=e^{i\theta}=e^{i(\theta+4\pi)}=...=e^{i(\theta+2k\pi)}, k\in\mathbb{Z}$
• Examples:
  • $e^{i\frac{\pi}{2}}=i,$
  • $e^{i\pi}=-1,$
  • $e^{i\frac{\pi}{4}}=\frac{1+i}{\sqrt{2}},$
  • $e^{2\pi i}=1,...$

Properties of the Exponential Notation

• $|e^{i\theta}|=1.$
• $\overline{e^{i\theta}}=e^{-i\theta}.$
• $\frac{1}{e^{-i\theta}}=e^{-i\theta}.$
• $e^{i(\theta+\varphi)}=e^{i\theta}\cdot e^{i\varphi}.$

Conclusions for the Argument Function

• $\text{arg}(\overline{z})=-\text{arg}z.$
• $\text{arg}\left(\frac{1}{z}\right)=-\text{arg}z.$
• $\text{arg}(z_1 z_2)=\text{arg}(z_1)+\text{arg}(z_2).$
• Examples:
  • $\text{arg}(i\cdot i)=\text{arg}(-1)=\pi, \text{arg}(i)+\text{arg}(i)=\frac{\pi}{2}+\frac{\pi}{2}=\pi.$
  • $\text{arg}((-1)(-1))=\text{arg}(1)=0,\text{arg}(-1)+\text{arg}(-1)=\pi+\pi=2\pi.$

Multiplication in Polar Form

• Consider $z_1=r_1e^{i\varphi_1}$ and $z_2=r_2e^{i\varphi_2}$. What is the polar form of $z_1 z_2?$
• $z_1 z_2=r_1 e^{i\varphi_1} r_2 e^{i\varphi_2}=(r_1 r_2)e^{i(\varphi_1+\varphi_2)}.$

De Moivre's Formula

• $e^{i\theta}\cdot e^{i\theta}=e^{i(\theta+\theta)}=e^{i\cdot2\theta}.$
• $\left(e^{i\theta}\right)^3=e^{i\cdot3\theta}.$
• $\left(e^{i\theta}\right)^n=e^{in\theta}$ (also true for negative $n$).
• Recall that $e^{i\theta}$ is simply short for $cos\ \theta+i\ sin\ \theta$. Thus this last formula means:
• $(cos\ \theta+ i\ sin\ \theta)^n=cos(n\theta)+i\ \sin(n\theta).$

Consequences of De Moivre's Formula

• $(cos\ \theta+i\ sin\ \theta)^n=cos(n\theta)+i\ \sin(n\theta).$
• This can be used to derive equations for sine and cosine. Ex: $n=3:$
  • $cos(3\theta)=cos^3\ \theta-3\ \cos\theta\ sin^2\theta.$
  • $sin(3\theta)=3\ cos^2\ \theta\ \sin\ \theta-sin^3\ \theta.$

The $n$th Root

• If $w\neq0$ there are exactly $n$ distinct $n$th roots:
• The polar form for $w$ and $z: w=\rho e^{i\varphi}$ and $z=r e^{i\theta}.$
• The equation $z^n=w$ then becomes

\[
r^ne^{i n\theta}=\rho e^{i\varphi} \text{, so } r^n=\rho \text{ and } e^{i n \theta}=e^{i\varphi}.
\]

• Thus $r = \sqrt[n]{\rho}$ and $n\theta=\varphi+2k\pi, k\in\mathbb{Z},$ so $\theta=\frac{\varphi}{n}+\frac{2k\pi}{n}, k=0,1,...,n-1.$
• We write $w^{\frac{1}{n}}=\sqrt[n]{\rho}e^{i\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)}, k=0,1,...,n-1.$

Examples $n$th Roots

\[
w=\rho e^{i\varphi}, w^{\frac{1}{n}}=\sqrt[n]\rho e^{i\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)}, k=0,1...,n-1.
\]

• Square roots of $4i:$

\[
4i=4e^{i\frac{\pi}{2}}, \text{ so }\rho=4, \varphi=\frac{\pi}{2}\text{ and }n=2.
\]

\begin{align*}
\left(4i\right)^{\frac{1}{2}}        &=\sqrt{4}\cdot e^{i\left(\frac{\pi}{4}+\frac{2k\pi}{2}\right)}, k=0, 1\\
&=\begin{cases}
2\cdot e^{i\frac{\pi}{4}}                                                                                          &\text{if }k=0\\
2\cdot e^{i\frac{\pi}{4}+\pi}                                                                                  &\text{if }k=1
&\end{cases}\\
&=\pm\left(\sqrt{2}+i\sqrt{2}\right).
\end{align*}

\[
w=\rho e^{i\varphi}, w^{\frac{1}{n}}=\sqrt[n]{\rho}e^{i\left(\frac{\rho}{n}+\frac{2k\pi}{n}\right)}, k=0,1,..., n-1.
\]

• Cubed roots of -8:

\[
-8 = 8e^{i\pi} \text{, so } \rho=8, \varphi=\pi \text{ and } n=3.
\]

\begin{align*}
\left(-8\right)^{\frac{1}{3}}        &=\sqrt[3]{8}\cdot e^{i\left(\frac{\pi}{3}+\frac{2k\pi}{3}\right)}, k=0, 1, 2\\
&=\begin{cases}
2\cdot e^{i\frac{\pi}{3}}                                                                                            &\text{if }k=0\\
2\cdot e^{i\pi} = -2                                                                                                      &\text{if }k=1\\
2\cdot e^{i\frac{5\pi}{3}}                                                                                          &\text{if }k=2.
&\end{cases}\\
\end{align*}

Roots of Unity

Since $1=1e^{i\cdot 0},$ we find that

\begin{align*}
1^{\frac{1}{n}}        &=\sqrt[n]{1}\cdot e^{i\left(\frac{0}{n}+\frac{2k\pi}{n}\right)}, &k=0, 1,..., n-1\\
&=e^{i\frac{2\pi k}{n}}, & k=0,1,..., n-1.
\end{align*}

Sets in the Complex Plane

• Circles and disks: center $z_0=x_0+iy_0$, radius $r.$
  • $B_r(z_0) = \{z \in \mathbb{C} : z$ has a distance less than $r$ from $z_0\}$ disk of radius $r$, centered at $z_0.$
  • $K_r(z_0) = \{z \in \mathbb{C} : z$ has a distance $r$ from $z_0\}$ circle of radius $r$, centered at $z_0.$
• \begin{align*}d\end{align*}
• Distance is measured by
  • \begin{align*}
    d &=\sqrt{(x - x_0)^2 + (y-y_0)^2}\\
    &=|(x - x_0) + i(y - y_0)|\\
    &=|z - z_0|.
    \end{align*}
• So $B_r(z_0) = \{z \in \mathbb{C} : |z - z_0| < r\}$ and $K_r(z_0) = \{z \in \mathbb{C} : |z - z_0| = r\}.$

Interior Points and Boundary Points

Open and Closed Sets

Examples:

• $\{z \in \mathbb{C} : |z - z_0| < r\}$ and $\{z \in \mathbb{C} : |z - z_0| > r\}$ are open.
• $\mathbb{C}$ and $\varnothing$ are open.
• $\{z \in \mathbb{C} : |z - z_0| \leq r\}$ and $\{z \in \mathbb{C} : |z - z_0| = r \}$ are closed.
• $\mathbb{C}$ and $\varnothing$ are closed.
• $\{z \in \mathbb{C} : |z - z_0| < r\} \cup \{z \in \mathbb{C} : |z - z_0| = r$ and $\Im(z - z_0) > 0\}$ is neither open nor closed.

Closure and Interior of a Set

Examples:

• $\overline{B_r(z_0)} = B_r(z_0) \cup K_r(z_0) = \{z \in \mathbb{C} : |z - z_0| \leq r \}.$
• $\overline{K_r(z_0)} = K_r(z_0).$
• $\overline{B_r(z_0) \backslash \{z_0\}} = \{z \in \mathbb{C} : |z - z_0| \leq r\}.$
• With $E = \{z \in \mathbb{C} : |z - z_0| \leq r\}, \overset{\circ}{E} = B_r(z_0).$
• With $E = K_r(z_0), \overset{\circ}{E} = \varnothing.$

Connectedness

Intuitively: A set is connected if it is 'in one piece'. How do we make this precise?

Example:

\[
X =  [0,1) \text{ and } Y = (1,2]
\]

are separated: For example, choose $U = B_1(0)$ and $V = B_1(2).$ Thus

\[
X \cup Y = [0, 2] \backslash \{1\}
\]

is not connected. It is hard to check whether a set is connected.

Connectedness for Open Sets in \mathbb{C}

For open sets, there is a much easier criterion to check whether or not a  set  is connected:

Bounded Sets

The Point at Infinity

• In $R,$ there are two directions which give rise to $\pm\infty.$
  \[
  1, 2 ,3 ,4 ,5 ,... \to \infty; -1, -2, -3, -4, -5,... \to -\infty.
  \]

• In $\mathbb{C},$ there is only one $\infty$ which can be attained in many directions.
  \[
  \left.\begin{aligned}
  &1, 2, 3, \ldots\\
  &-1, -2, -3, \ldots\\
  &i, 2i, 3i, \ldots\\
  &1, 2i, -3, -4i, 5, 6i, -7, \ldots\\
  &\vdots\\
  \end{aligned}\right\} \to\infty
  \]