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I haven't been able to find whether or not De Moivre's Theorem ($(\cos\theta +i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ ) is true for all real numbers n or if it's just true for integer values of n (I'm only taught the latter).

If not, is there a function that makes it true for all real n, or at least all rational n?

Thanks

frog1944
  • 2,357
  • It's certainly true for all integers, but if $n$ becomes fractional, great care must be taken into account. For example: $z^2=i$ has two solutions but you are not going to get both of them if you only consider $n=\frac{1}{2}$ with DeMoivre. – imranfat Nov 12 '16 at 03:07
  • If you take $n=1/2$ and $\theta=0$, then $\theta=2\pi$, you'll obtain two different values; so the formula fails. – MattG88 Nov 12 '16 at 03:09
  • https://en.m.wikipedia.org/wiki/Exponentiation#Powers_of_complex_numbers – lab bhattacharjee Nov 12 '16 at 07:25

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