# Complex Exponentials and Trig Identities

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