De Moivre’s theorem

If \(z=r(\cos\theta+i\sin\theta)\), then \(z^n=r^n(\cos n\theta+i\sin n\theta)\) for positive integers \(n\). For \(z\ne0\), the formula also holds for zero and negative integer powers.

Example

\(\left(2\mathrm e^{\frac{i\pi}{3}}\right)^4=16\mathrm e^{\frac{4\pi i}{3}}=-8-8\sqrt3\,i\).

Proof for positive integers

Let \(P(n)\) be \((\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta\). The case \(n=1\) is immediate.

Assume \(P(k)\) for some positive integer \(k\). Using the polar multiplication rule,

\(\begin{aligned}(\cos\theta+i\sin\theta)^{k+1}&=(\cos k\theta+i\sin k\theta)(\cos\theta+i\sin\theta)\\&=\cos((k+1)\theta)+i\sin((k+1)\theta).\end{aligned}\)

Thus \(P(k+1)\) follows from \(P(k)\). By induction, the result holds for all positive integers \(n\); multiplying by \(r^n\) gives the general form.

Rational powers require all possible arguments before dividing an angle. The roots method in the next section supplies all values; taking only a principal angle can miss roots.

For a unit complex number \(\omega=\mathrm e^{i\theta}\), each multiplication by \(\omega\) rotates through \(\theta\). Thus \(\omega^n=\mathrm e^{in\theta}\) represents \(n\) such rotations when \(n\) is a positive integer.