Theory

Theory 1

Multiplication of complex numbers is much easier to understand when the numbers are written using polar form.

There is a shorthand ‘’ notation. Convert to polar coordinates, so and :

The “” stands for . For example:

---

Euler Formula

General Euler Formula:

On the unit circle :

The form expresses the same data as the form. The principal advantage of the form is that it reveals the rule for multiplication, which comes from exponent laws:

Complex multiplication - Exponential form

In words:

• Multiply radii
• Add angles

Notice:

Notice:

Therefore ‘acts upon’ other numbers by rotating them counterclockwise!

---

De Moivre’s Theorem - Complex powers

In exponential notation:

In notation:

Expanded notation:

So the power of acts like this:

• Stretch: to
• Rotate: by increments of

---

Extra - Derivation of Euler Formula

Recall the power series for :

Plug in :

Simplify terms:

Separate by -factor. Select out the :

Separate into a series without and a series with :

Identify and . Write trig series:

Therefore .