Powers of Complex Numbers (De Moivre’s Theorem)

Powers of Complex Numbers (De Moivre’s Theorem)

Introduction to De Moivre’s Theorem

De Moivre’s Theorem is a fundamental result in trigonometry and complex numbers, allowing us to compute powers and roots of complex numbers expressed in polar form. This theorem is especially useful for simplifying expressions involving complex numbers raised to integer powers.

• Complex Number in Polar Form: Any complex number can be written as z = r(cosb8 + i
  1sinb8), where r is the modulus and b8 is the argument.

• De Moivre's Theorem: For any integer n,

$[r(
1cos\theta + i\sin\theta)]^n = r^n [\cos(n\theta) + i\sin(n\theta)]$

Product and Power of Complex Numbers in Polar Form

• Product: If z_1 = r_1(
  1cos\theta_1 + i\sin\theta_1) and z_2 = r_2(
  1cos\theta_2 + i\sin\theta_2), then:

$z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$

• Power: For z = r(
  1cos\theta + i\sin\theta),

$z^n = r^n [\cos(n\theta) + i\sin(n\theta)]$

Example: Calculating a Power of a Complex Number

• Given: $|4\text{cis}\frac{2\pi}{3}|^3$

• Solution:

$[4\text{cis}\frac{2\pi}{3}]^3 = 4^3 \text{cis}(3 \times \frac{2\pi}{3})$

$= 64\text{cis}2\pi$

• Where: $\text{cis}\theta = \cos\theta + i\sin\theta$

Summary Table: Product and Power of Complex Numbers

Form	Product	Power
$z_1 = r_1\text{cis}\theta_1$ $z_2 = r_2\text{cis}\theta_2$	$z_1 z_2 = r_1 r_2 \text{cis}(\theta_1 + \theta_2)$	$z^n = r^n \text{cis}(n\theta)$

Key Points

• De Moivre’s Theorem simplifies raising complex numbers in polar form to integer powers.

• cis notation is shorthand for $\cos\theta + i\sin\theta$.

• Multiplying complex numbers in polar form involves multiplying their moduli and adding their arguments.

Additional info: De Moivre’s Theorem is also used to find roots of complex numbers and is foundational for understanding trigonometric identities and their applications in complex analysis.