February 28, 2020
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.
with a modulus
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.
We'll extend the result for and apply it to
If with then
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 .
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
The Real part of the expansion is
and the Imaginary part of the expansion is