1 De Moivre’s theorem
We have seen, in Section 10.2 Key Point 7, that, in polar form, if and then the product is:
In particular, if and (i.e. ), we obtain
Multiplying each side of the above equation by gives
on adding the arguments of the terms in the product.
Similarly
After completing such products we have:
where is a positive integer.
In fact this result can be shown to be true for those cases in which is a negative integer and even when is a rational number e.g. .
Recalling from Key Point 8 that , De Moivre’s theorem is simply a statement of the laws of indices: