Usefulness of De Moivre's Theorem

Date: 2/10/96 at 10:9:18
From: Anonymous
Subject: Use of DeMoivre's Theorem

What is the usefulness of DeMoivre's theorem?

Date: 2/10/96 at 12:18:0
From: Doctor Sarah
Subject: Re: Use of DeMoivre's Theorem

Hello!

De Moivre's theorem involves the branch of trigonometry that
deals with imaginary quantities.

Here's some information found by doing a Web search on de Moivre.  It's 
from a Web page called The English Mathematicians of the Eighteenth 
Century: 

_A Short Account of the History of Mathematics_ (4th edition, 1908) by 
W. W. Rouse Ball.  The URL is:

http://www.maths.tcd.ie/pub/HistMath/People/18thCentury/RouseBall/RB_Engl18C.html   

"He is best known for having, together with Lambert, created 
that part of trigonometry which deals with imaginary quantities. 
Two theorems on this part of the subject are still connected with 
his name, namely, that which asserts that sin nx + i cos nx is one 
of the values of (sin x + i cos x)^n, and that which gives the 
various quadratic factors of x^(2n) - 2p x^n + 1."

Here's something about de Moivre's life:

"Demoivre (more correctly written as de Moivre) was born at 
Vitry on May 26, 1667, and died in London on November 27, 
1754. His parents came to England when he was a boy, and his 
education and friends were alike English. His interest in the 
higher mathematics is said to have originated in his coming by 
chance across a copy of Newton's _Principia_. From the eulogy 
on him delivered in 1754 before the French Academy it would 
seem that his work as a teacher of mathematics had led him to the 
house of the Earl of Devonshire at the instant when Newton, who 
had asked permission to present a copy of his work to the earl, 
was coming out. Taking up the book, and charmed by the far-
reaching conclusions and the apparent simplicity of the 
reasoning, Demoivre thought nothing would be easier than to 
master the subject, but to his surprise found that to follow the 
argument overtaxed his powers. He, however, bought a copy, 
and as he had but little leisure he tore out the pages in order to 
carry one or two of them loose in his pocket so that he could 
study them in the intervals of his work as a teacher. 
Subsequently he joined the Royal Society, and became intimately 
connected with Newton, Halley, and other mathematicians of the 
English school. The manner of his death has a certain interest 
for psychologists. Shortly before it he declared that it was 
necessary for him to sleep some ten minutes or a quarter of an 
hour longer each day than the preceding one. The day after he 
had thus reached a total of something over twenty-three hours he 
slept up to the limit of twenty-four hours, and then died in his 
sleep.

"His chief works, other than numerous papers in the 
_Philosophical Transactions_, were _The Doctrine of Chances_, 
published in 1718, and the _Miscellanea Analytica_, published in 
1730. In the former the theory of recurring series was first 
given, and the theory of partial fractions which Cotes's 
premature death had left unfinished was completed, while the 
rule for finding the probability of a compound event was 
enunciated. The latter book, besides the trigonometrical 
propositions mentioned above, contains some theorems in 
astronomy, but they are treated as problems in analysis."

-Doctor Sarah,  The Math Forum