de Moivre's Theorem
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 forand 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 ofandrespectively.
De Moivre's theorem can be used to expressorin terms of powers oforrespectively. To do this we replacewith(the difference between an expression in terms oforis trivial), obtaining
Suppose then we want to find expressions forand
Now separate the real and imaginary components on both sides, obtaining the two equations
Substituteinto the first of these two:
Substituteinto the second equation: