Chapters

Powers of Complex Numbers Introduction

We can find powers of Complex numbers, like , by either performing the multiplication by hand or by using the Binomial Theorem for expansion of a binomial . This can be somewhat of a laborious task. Fortunately, there is a nifty shortcut that we can apply to shorten the process and it involves the Polar form of Complex numbers.

Example

Find

with a modulus

which equals

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Powers of the Polar Form

We saw in the Polar Representation section the proof that

and we're going to extend this definition to show that the power of any Complex number also has a very special and useful result.

Example

We'll extend the result for and apply it to

If with then

then

Example

If with then

We can now infer that

De Moivre's Theorem and The Unit Circle

By setting and using the Unit Circle, we obtain De Moivre's Theorem

This is an extremely useful theorem for finding powers and roots of Complex numbers.

We can express the sine or cosine function of a multiple of  an angle , , by powers of the sine and cosine of the original angle . We do this by applying the Binomial Theorem for a power to the product .

Example

If then we can expand the right side in powers of the sine and cosine of by using the binomial expansion for

Binomial Expansion for :

for decreasing powers of cosine from and increasing powers of sine from

then

The Real part of the expansion is

and the Imaginary part of the expansion is

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