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De Moivre's Theorem

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DeMoivre's Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for x\in\mathbb{R} and n\in\mathbb{Z}, \left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx).

Proof

This is one proof of De Moivre's theorem by induction.

  • If n>0, for n=1, the case is obviously true.
Assume true for the case n=k. Now, the case of n=k+1:
Image:DeMoivreInductionP1.gif
Therefore, the result is true for all positive integers n.
  • If n=0, the formula holds true because \cos(0x)+i\sin (0x)=1+i0=1. Since z^0=1, the equation holds true.
  • If n<0, one must consider n=-m when m is a positive integer.
Image:DeMoivreInductionP2.gif

And thus, the formula proves true for all integral values of n. \Box

Note that from the functional equation f(x)^n = f(nx) where f(x) = \cos x + i\sin x, we see that f(x) behaves like an exponential function. Indeed, Euler's formula states that e^{ix} = \cos x+i\sin x\right. This extends De Moivre's theorem to all n\in \mathbb{R}.

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