Polar form and De Moivre's theorem

Polar (mod-arg) form

• A complex number can be written using its modulus $r$ and argument $\theta$ instead of its real and imaginary parts:

$z = r(\cos\theta + i\sin\theta) = r\,\text{cis}\,\theta$

• "$\text{cis}\,\theta$" is shorthand for "$\cos\theta + i\sin\theta$".
• Rectangular → polar: find $r = |z|$ and $\theta = \arg(z)$.
• Polar → rectangular: work out $a = r\cos\theta$ and $b = r\sin\theta$.

Multiplying and dividing in polar form

• Polar form makes multiplication and division easy:
  • Multiply: multiply the moduli and add the arguments.
  • Divide: divide the moduli and subtract the arguments.

$r_1\text{cis}\,\theta_1 \times r_2\text{cis}\,\theta_2 = r_1 r_2\,\text{cis}(\theta_1 + \theta_2)$

De Moivre's theorem

• De Moivre's theorem raises a complex number in polar form to a power:

$\left[r\,\text{cis}\,\theta\right]^n = r^n\,\text{cis}(n\theta)$

• Raise the modulus to the power and multiply the argument by $n$. This turns an otherwise huge expansion into two quick steps.