DeMoivre's Formula

Date: 08/13/98 at 05:18:23
From: Greg Lukens
Subject: Demove's Formula

I have forgotten about DeMoivre's formula. Thanks in advance for 
the reminder.

Greg Lukens

Date: 08/13/98 at 14:07:29
From: Doctor Benway
Subject: Re: Demove's formula

Hi Greg,

I believe the formula you're looking for is DeMoivre's formula, which 
is the following:

   (cos(theta) + i*sin(theta))^n = cos(n*theta) + i*sin(n*theta)

This formula is useful when you have a complex number and want to raise 
it to some power without doing a lot of work.

If all you want is the formula, you can ignore the rest of this 
message. However, if you want a little more insight into what is going 
on, read on.

Recall that any complex number can be written in the form 
r*e^(i*theta). If you plot a complex number in the complex plane 
(where the x-axis is the real axis and y-axis is the imaginary axis), 
then "r" will be the distance from the point to the origin and theta 
will be the angle a line from the origin to the point makes with the 
x-axis. A little trig shows that a complex number written as 
r*e^(i*theta) can also be written as r*cos(theta)+r*i*sin(theta). 

Knowing this little fact gives us the ability to switch back and forth 
between ways of writing complex numbers, depending on what we want to 
do with them. If we want to add complex numbers, then the form a + b*i 
is easiest, whereas if we want to multiply them together, it is easier 
to use the form r*e^(i*theta).  

Essentially what you are doing is taking a complex number of the form 
(a + b*i), converting it to the form r*e^(i*theta), raising it to a 
power in that form, then converting back to the first form.  Observe:

   (r*cos(theta) + r*i*sin(theta))^n
      = (r*(cos(theta) + i*sin(theta))^n
      = (r^n) * (cos(theta) + i*sin(theta))^n
      = (r^n) * (e^(i*theta))^n
      = (r^n) * (e^(n*i*theta))
      = (r^n) * (cos(n*theta) + i*sin(n*theta))

Of course knowing DeMoivre's formula allows us to go straight from 

   (r*(cos(theta) + i*sin(theta))^n 

to 

   (r^n) * (cos(n*theta) + i*sin(n*theta)).

Thanks for writing, hope this helps.  

- Doctor Benway, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 08/13/98 at 16:55:10
From: Greg Lukens
Subject: Re: Demove's formula

Hi Dr. Math,

That was way more than I was looking for. I found it quite interesting 
and am glad to know it.

Thanks for your help,
Greg