# Polar form.

We denote by the set of all complex numbers whose absolute value is equal to 1.

For example, and , but .

The image of in the Cauchy-Argand plane is the unit circle: if , with real , then if, and only if .

Proposition 1.3.1   Let ; then exists and is an element of .

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

Proposition 1.3.2   For any and , we have .

Proof. For any and , we have:

whence .

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

Theorem 1.3.3   Let . There exists a real number such that .

Proof. Denote , where and are real numbers. Then if, and only if, , i.e. the image of in the complex plane is a point on the unit-circle. For each point on the unit-circle, there exist a real number such that the coordinates of this point are , whence the result.

Definition 1.3.4   The number is called an argument of and is denoted .

Note that this argument is defined up to an additional .

Example 1.3.5

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

We can now generalize this to any non zero complex number. First note that for any , we have , as the following holds:

Definition 1.3.6   For any ,

Example 1.3.7
1. Take . The absolute value of is given by . Then we have:

It follows that .
2. Now take , then and the following hold:

It follows that .

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)
.
(ii)
is pure imaginary .

Proof. Let ; denote . Then:
(i)
.
(ii)