De Moivre's Theorem relates a complex number to it's polar form. It states
Proof: We can prove De Moivre's Theorem using Taylor series.
We separate the Taylor series into real and complex parts. The real part is the Taylor series for
and the imaginary part is the Taylor series for
This becomes (1) on recognizing that on the right hand side we have expressions for the Taylor series of
and
respectively.
De Moivre's theorem can be used to express
or
in terms of powers ofor
respectively. To do this we replace
with
(the difference between an expression in terms of
or
is trivial), obtaining
But
hence
(2)
Suppose then we want to find expressions for
and
Now separate the real and imaginary components on both sides, obtaining the two equations
and
Substitute
into the first of these two:
ie
Substitute
into the second equation:
