Complex Numbers and De Moivre's Theorem

Complex Numbers and De Moivre's Theorem

Polar Form

• A complex number $z = x + iy$ can be written in polar form as: $$z = r(\cos\theta + i\sin\theta)$$ where $r = |z| = \sqrt{x^2 + y^2}$ and $\theta = \arg(z)$.
• Exponential form: $z = re^{i\theta}$, using Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$.

De Moivre's Theorem

• For any integer $n$: $$(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$$
• Also: $(e^{i\theta})^n = e^{in\theta}$.

Roots of Unity

• The $n$th roots of unity are solutions to $z^n = 1$: $$z_k = e^{2\pi i k / n} = \cos\left(\frac{2\pi k}{n}\right) + i\sin\left(\frac{2\pi k}{n}\right), \quad k = 0, 1, \dots, n-1$$
• They lie on the unit circle, equally spaced by $\frac{2\pi}{n}$.

Cube Roots

• For a complex number $z = r(\cos\theta + i\sin\theta)$, the cube roots are: $$w_k = \sqrt[3]{r} \left( \cos\left(\frac{\theta + 2\pi k}{3}\right) + i\sin\left(\frac{\theta + 2\pi k}{3}\right) \right), \quad k = 0, 1, 2$$

Applications

• Powers of trigonometric functions: Using De Moivre's theorem, $\cos(nx)$ and $\sin(nx)$ can be expressed as polynomials in $\cos x$ and $\sin x$.
• Example: $\cos(5\theta) = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta$.

Key Identities

• $e^{ix} + e^{-ix} = 2\cos x$ and $e^{ix} - e^{-ix} = 2i\sin x$.
• Hence, $e^{inx} + e^{-inx} = 2\cos(nx)$ and $e^{inx} - e^{-inx} = 2i\sin(nx)$.