# MATH3401

Lecture 3

## Complex conjugate

$$\conj{\cdot} : \C \ra \C$$, $$x+i\,y \mapsto x-i\,y$$.

If $$z=x+i\,y$$, then $\overline{z} = x-i\,y.$

## Complex conjugate

Properties

(i) $$z = \conj{z} \leftrightarrow \Im(z)=0$$, i. e. $$z\in \mathbb R$$.

(ii) $$\conj{\left( \conj{z} \right)} = z$$.

(iii) $$\conj{\left( z \cdot w \right)} = \conj{z} \cdot \conj{w}$$.

(iv) $$\conj{\left( \dfrac{1}{z} \right)} = \dfrac{1}{\conj{z}}, \; z\neq 0$$.

## Complex conjugate

Properties

(v) $$|z|^2 = z \cdot \conj{z}$$.

(vi) $$\Re(z) = \dfrac{z+\conj{z}}{2}$$, $$\Im(z) = \dfrac{z-\conj{z}}{2i}$$.

(vii) $$\conj{z + w} = \conj{z} +\conj{w}$$.

### Triangle inequality

$$|z+w|\leq |z| + |w|$$

Proof:

$$|z+w|^2=|z|^2+|w|^2-2|z||w|\cos A$$

$$|z+w|^2\leq |z|^2+|w|^2+2|z||w|$$,
since $$\cos A \geq -1$$.

Then $$|z+w|^2 \leq \left(|z|+|w|\right)^2$$.

Take square root of both sides and we are done $$\blacksquare$$.

### Polar coordinates

$$\left\{ \begin{array}{r} x = r\cos \theta \\ y = r\sin \theta \end{array} \right.$$

\begin{align*} z &= r e^{i \theta} \\ &= r \left(\cos \theta + i \, \sin \theta \right)\\ &= r \,\text{c}i\text{s}\,\theta \end{align*}

### Argument

$$r e^{i \theta}$$ formally follows from Taylor series of $$e^{i\theta}$$.

Here $$\theta$$ is an argument of $$z$$ and we write $$\theta = \arg(z)$$.

The $$\arg$$ is not a (single-valued) function.

If $$\theta$$ is an argument of $$z$$, $$\Arg(z)$$ is defined to be the unique value of $$\theta$$ with $-\pi \lt \arg (z) \leq \pi$

### Argument

$$\Arg(1+i)=\frac{\pi}{4}$$, but $\arg(1+i) = \ldots, -\frac{7\pi}{4}, \frac{\pi}{4}, \frac{9\pi}{4}, \ldots.$

$$\Arg(-6)=\pi$$.

$$\Arg(0)$$ is undefined.

$$\arg(0)=\R$$.

### Argument

So $$\Arg$$ is a function: $\C\setminus \{0\} \ra (-\pi, \pi]$

$$\C\setminus \{0\} = \C^* = \C_*$$.

### Argument

Notice also that $$|e^{i\theta}|=1$$ 📝 (← This symbol means 'check it').

$$\left(e^{i\theta} \right)^{-1} = e^{-i\theta}= \conj{e^{i\theta}}$$.

$$\left( r e^{i\theta} \right) \left( \rho e^{i\phi} \right) = r \rho e^{i\left(\theta + \phi\right)}$$.

So $$|z\, w| = |z|\,|w|$$ and $\arg(z\, w) = \arg(z) + \arg(w).$

### Argument

But $$\Arg(z\, w)$$ may fail to be equal to $\Arg(z)+\Arg(w).$

E. g. $$z = w = \dfrac{-1+i}{2}$$.

$$\Arg(z) = \Arg(w) = \dfrac{3\pi}{4}$$.

$$z\,w = -i \implies \Arg(z \, w) = -\dfrac{\pi}{2}$$.

but $$\Arg(z)+\Arg(w) = \dfrac{3\pi}{2}$$.

### de Moivre's formula

Let $$z = r e^{i\theta}$$, then $$z^n = r^n e^{i n \theta}$$, $$n\in \N$$.

For $$r=1$$, \begin{align*} e^{i n \theta} &= \left(\cos \theta + \sin \theta \right)^n \\ &= \cos \left(n\theta\right) + i\,\sin \left(n\theta\right) \end{align*} de Moivre's formula.

Example: \begin{align*} \left(1+i\right)^7 &= \left( \sqrt{2} e^{i\pi/4} \right)^7 \\ &= \left(\sqrt{2}\right)^7 e^{7\pi i/4} = 8-8i. \end{align*}