Recall that
The angles add. You've seen something similar before:
This connection between exponentiation and (4.4)
gives us an idea!
If is a complex number, *define*

We have just written polar coordinates in another form. It's
a shorthand for the polar form of a complex number:

**Theorem 4.4.1**
*If , are two complex numbers, then
*
*Proof*.

Here we have just used (

4.4).

The following theorem is amazing, since it involves calculus.

**Theorem 4.4.2**
*If is a complex number, then
*
*
for real. In fact, this is even true for
a complex variable (but we haven't defined differentiation
for complex variables yet). *
*Proof*.
Write

.

Now we use the product rule to get

On the other hand,

Wow!! We did it!

That Theorem 4.4.2 is true is pretty amazing. It's
what really gets complex analysis going.

**Example 4.4.3**
Here's another fun fact:

*Solution.* By definition,
have

.

**Subsections**
William Stein
2006-03-15