about De Moivre’s Theorem

De Moivre's Theorem
As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem:


圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img217.png


Working this out using sum-of-angle identities from trigonometry is laborious (see §3.13 for details). However, using Euler's identity, De Moivre's theorem simply ``falls out'':


圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img218.png


Moreover, by the power of the method used to show the result,
圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img69.png
can be any real number, not just an integer.
so if you using Euler's identity, the result is suitable for n is an irrational number is not difficult to see