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:

The notation stands for .

For example:

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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:

Complex multiplication - Exponential form

In words:

• Multiply radii
• Add angles

Notice:

Notice:

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

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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

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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 .