Polar Form, Euler Form, and De Moivre’s Theorem

Complex numbers can be expressed in various forms, including Cartesian form, polar form, and Euler form. These representations allow for efficient computation of sums, products, quotients, powers, and roots of complex numbers.

Key Concepts

• Modulus–Argument (Polar) Form:
  • The polar form of a complex number is \( z = r(\cos\theta + i\sin\theta) \), also written as \( z = r\text{cis}\theta \), where:
    • \( r = |z| = \sqrt{a^2 + b^2} \): modulus (magnitude).
    • \( \theta = \text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \): argument (angle).
• Euler Form:
  • Using Euler’s formula, \( e^{i\theta} = \cos\theta + i\sin\theta \), the polar form can be rewritten as:
    • \( z = re^{i\theta} \).
• Conversions Between Forms:
  • From Cartesian to Polar/Euler:
    • Find \( r = \sqrt{a^2 + b^2} \) and \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
    • Express as \( z = r\text{cis}\theta \) or \( z = re^{i\theta} \).
  • From Polar/Euler to Cartesian:
    • Expand \( z = r(\cos\theta + i\sin\theta) \) or \( z = re^{i\theta} \) to get \( z = a + bi \).
• Sums, Products, and Quotients:
  • Geometric interpretation is important for understanding operations on complex numbers in polar or Euler form:
    • Product: \( z_1 \cdot z_2 = r_1r_2\text{cis}(\theta_1 + \theta_2) \).
    • Quotient: \( \frac{z_1}{z_2} = \frac{r_1}{r_2}\text{cis}(\theta_1 – \theta_2) \).
• Complex Conjugates:
  • Complex conjugate roots occur in pairs for polynomials with real coefficients.
• De Moivre’s Theorem:
  • For \( z = r\text{cis}\theta \) and \( n \in \mathbb{Z}^+ \), \( z^n = r^n\text{cis}(n\theta) \).
  • Extension to rational exponents: \( z^{1/n} = r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right), k = 0, 1, \ldots, n-1 \).

Guidance, Clarifications: 

• Proof of De Moivre’s theorem for \( n \in \mathbb{Z}^+ \) is expected using mathematical induction.
• Applications include:
  • Finding powers and roots of complex numbers.
  • Link to compound angle identities: \( (\cos\theta + i\sin\theta)^n \) expands to trigonometric expressions.
• Connections to polynomial equations:
  • Roots of a polynomial can be found using \( z^{1/n} \), and the sum and product of roots relate to coefficients (AHL 2.12).

Examples

Example 1: Convert \( z = 1 + \sqrt{3}i \) to Polar Form

• Modulus: \( r = \sqrt{1^2 + (\sqrt{3})^2} = 2 \).
• Argument: \( \theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3} \).
• Polar Form: \( z = 2\text{cis}\frac{\pi}{3} \) or \( z = 2e^{i\pi/3} \).

Example 2: Use De Moivre’s Theorem to Find \( (1 + i)^4 \)

• Convert to polar form: \( z = \sqrt{2}\text{cis}\frac{\pi}{4} \).
• Using De Moivre’s theorem: \( z^4 = (\sqrt{2})^4\text{cis}(4 \cdot \frac{\pi}{4}) = 4\text{cis}\pi = -4 \).

Example 3: Roots of \( z^3 = 8 \)

• Convert \( z \) to polar form: \( z = 8\text{cis}0 \).
• Roots: \( z^{1/3} = 2\text{cis}\left(\frac{0 + 2k\pi}{3}\right), k = 0, 1, 2 \).
• Roots: \( z_0 = 2, z_1 = 2\text{cis}\frac{2\pi}{3}, z_2 = 2\text{cis}\frac{4\pi}{3} \).