chapter1

Imaginary unit: $i = sqrt{-1}$

We can see that:

$$i^2 = (sqrt{-1})^2 = -1, quad i^3 = i^2cdot i = -i,quad i^{4} = (-1) cdot (-1) = 1$$

Complex number: number of the form $z = a + bi​$, where $a, b in mathbb{R}​$. The set of complex number will be denoted by $mathbb{C}​$.

$a$: real part

$b$: imaginary part

$mathbb{R} in mathbb{C}$

Use Complex plane to envision complex numbers.

Figure 1.1 Some complex numbers in the complex plane

Define two complex numbers:

$$u = a + bi, quad v = c + d i \$$ $$begin{eqnarray*} text{Addition: } &u + v &=& (a + c) + (b + d)i \ text{Subtraction: } &u -v &=& (a - c) + (b - d)i \ text{Multiplication:} &ucdot v &=& (a + bi) cdot(c+ di) \ & &=& ac + adi + bci + bdi^2 \ & &=& (ac-bd) + (ad + bc) i end{eqnarray*}$$

Example 1.1 (Complex Arithmetic) —> Page 2

Definition 1.1 (Conjugation of a Complex Number): Let $z = a + bi in mathbb{C}$ . The conjugation of $z$ , denoted by $bar{z}$ , is defined by

$$bar{z} = a - bi$$

Geometrically speaking, the conjugate $bar{z}$ of $z$ is simply the reflection of $z$ over the real axis.

Figure 1.2 Complex numbers and their conjugations in the complex plane

Proposition 1.1 (Properties of the Conjugation Operator): Let $z = a + bi$ be a complex number. Then

$$begin{align*} text{(a)}quad& bar{bar{z}} = z\ text{(b)}quad& z in mathbb{R} quadtext{if and only if } bar{z} = z end{align*}$$

Definition 1.2 (Modulus of a Complex Number): The modulus of the complex number $z = a + bi$ is denoted by $|z|$ and is defined as

$$|z| = sqrt{a^2 + b^2}$$

Other names: length, absolute value, norm of $z$.

A natural relationship between $|z|$ and $bar{z}$ :

$${|z|}^2 = z cdotbar{z}$$

Given

$$z = a + bi\ y = c+ di\ y,z ne 0$$

How to express the quotient $z/y$ as a complex number?

$$frac{z}{y} = frac{a + bi}{c+ di} = frac{a + bi}{c + di} cdotfrac{c-di}{c-di} = frac{(ac+bd) + (bc-ad)i}{c^2 + d^2} = frac{ac+bd}{c^2 + d^2} + frac{bc - ad}{c^2 + d^2}i$$

Let $z = a + bi$ and $y = c + d i$, we have

$$begin{align*} & text{(a)}quad overline{y cdot z} = bar{y}cdotbar{z} \ & text{(b)}quad |z| = |bar{z}| \ & text{(c)}quad |y cdot z| = |y|cdot|z| \ & text{(d)}quad overline{y + z} = bar{y} + bar{z} \ & text{(e)}quad z^{-1} = frac{1}{z} = frac{1}{a+ bi} = frac{a}{|z|} -frac{b}{|z|}i \ & text{(d)}quad bar{z} = z^{-1}quadtext{if}quad|z| = 1 \ end{align*}$$

• suppose that $z, w in mathbb{C}$ with $|z|= 1$. Thus we have

$$|(z-w)(z-1/bar{w})| = |w|^{-1}|z-w|^{2}$$