Polar form.

We denote by $\mathbb{U}$ the set of all complex numbers whose absolute value is equal to 1.

$$\mathbb{U} = \{ z \in \mathbb{C} \; \vert \; \vert z\vert=1 \}$$

For example, $i \in \mathbb{U}$ and $\frac 45 + \frac 34 i \in \mathbb{U}$ , but $1+i \not\in \mathbb{U}$ .

The image of $\mathbb{U}$ in the Cauchy-Argand plane is the unit circle: if $z=x+iy$ , with real $x,y$ , then $z \in \mathbb{U}$ if, and only if $x^2+y^2=1$ .

Proposition 1.3.1   Let $z \in \mathbb{U}$ ; then $z^{-1}$ exists and is an element of $\mathbb{U}$ .

We leave the proof to the reader; use Corollary 1.18.

Proposition 1.3.2   For any $z_1 \in \mathbb{U}$ and $z_2 \in \mathbb{U}$ , we have $z_1z_2 \in \mathbb{U}$ .

Proof. For any $z_1 \in \mathbb{U}$ and $z_2 \in \mathbb{U}$ , we have:

$$\vert z_1z_2\vert=\vert z_1\vert \; \vert z_2\vert=1 \cdot 1=1,$$

whence $z_1z_2 \in \mathbb{U}$ . $\qedsymbol$

And now we will see how complex numbers are tied with trigonometry.

Theorem 1.3.3   Let $z \in \mathbb{U}$ . There exists a real number $\theta$ such that $z= \cos \theta + i \sin \theta$ .

Proof. Denote $z=x+iy$ , where $x$ and $y$ are real numbers. Then $z \in \mathbb{U}$ if, and only if, $x^2+y^2=1$ , i.e. the image of $z$ in the complex plane is a point on the unit-circle. For each point on the unit-circle, there exist a real number $\theta$ such that the coordinates of this point are $(\cos \theta, \; \sin \theta )$ , whence the result. $\qedsymbol$

Definition 1.3.4   The number $\theta$ is called an argument of $z$ and is denoted $\operatorname{arg}\nolimits (z)$ .

Note that this argument is defined up to an additional $2k \pi, k \in \mathbb{Z}$ .

Example 1.3.5  

$$\frac 12 + i \frac {\sqrt{3}}{2} = \cos \frac {\pi }{3} + i \sin \frac {\pi }{3} \Longrightarrow \... ...2 + i \frac {\sqrt{3}}{2} \right) = \frac {\pi }{3} + 2k \pi, k \in \mathbb{Z}.$$

$$i = \cos \frac {\pi }{2} + i \sin \frac {\pi }{2} \Longrightarrow \operatorname{arg}\nolimits (i) = \frac {\pi }{2} + 2k \pi, k \in \mathbb{Z}.$$

In Fig 5, a value of the argument of the complex number corresponding to a point is diplayed in green.

Figure 5: The unit circle in Cauchy-Argand plane.

We can now generalize this to any non zero complex number. First note that for any $z \in \mathbb{C}^*$ , we have $\frac {z}{\vert z\vert} \in \mathbb{U}$ , as the following holds:

$$\lvert \frac {z}{\vert z\vert} \rvert = \frac {\lvert z\rvert }{\lvert z\rvert }=1.$$

Definition 1.3.6   For any $z \in \mathbb{C}^*$ ,

$$\operatorname{arg}\nolimits (z)=\operatorname{arg}\nolimits \left(\frac {z}{\vert z\vert} \right).$$

Figure 6: The trigonometric form of a non zero complex number.

Example 1.3.7  

1. Take $z=1+i$ . The absolute value of $z$ is given by $\lvert z\rvert =\sqrt{1^1+1^2}=\sqrt{2}$ . Then we have:

   $$z= \sqrt{2}\; \left( \frac {1}{\sqrt{2}}+i \; \frac {1}{\sqrt{2}} \right) = \sqrt{2}\; ( \cos \frac {\pi}{4} + i \; \sin \frac {\pi}{4}).$$

   
   
   It follows that $\operatorname{arg}\nolimits (z)=\frac {\pi}{4} + 2k \pi$ .
2. Now take $z=-1+i \sqrt{3}$ , then $\lvert z\rvert =\sqrt{1+3}=2$ and the following hold:

   $$z=2 \; \left( - \frac 12 + i \; \frac{\sqrt{3}}{2} \right)= 2 \left( \cos \frac {2 \pi}{3} + i \; \sin \frac {2 \pi}{3} \right).$$

   
   
   It follows that $\operatorname{arg}\nolimits (z)= \frac {2 \pi}{3} + 2k \pi$ .

The following proposition is easy to understand. It explains the specific role of the coordinate axes when representing complex numbers in the Cauchy-Argand plane.

Proposition 1.3.8  

(i)

$z \in \mathbb{R} \Leftrightarrow \operatorname{arg}\nolimits (z)=k \pi, \; k \in \mathbb{Z}$ .

(ii)

$z$ is pure imaginary $\Leftrightarrow \operatorname{arg}\nolimits (z)= \frac {\pi}{2} +k \pi, \; k \in \mathbb{Z}$ .

Figure 7: Particular cases.

Proof. Let $z \in \mathbb{C}^*$ ; denote $\operatorname{arg}\nolimits (z)= \theta + 2k \pi, \; k \in \mathbb{Z}$ . Then:

(i)

$z \in \mathbb{R} \Leftrightarrow \operatorname{Im}\nolimits (z) =0 \Leftrighta... ...t \sin \theta =0 \Leftrightarrow \sin \theta =0 \Leftrightarrow \theta = 2k \pi$ .

(ii)

Please do it yourself.

$\qedsymbol$