Euler's formula: e^(itheta) = cos(theta) + isin(theta)

Polar form: z = re^(itheta) where r = |z| and theta = arg(z)

Why polar form is powerful:

• Multiplication: z1z2 = r1r2 * e^(i(theta1+theta2)) -- multiply moduli, add arguments
• Division: z1/z2 = $\frac{r1}{r2}$ * e^(i(theta1-theta2))
• Powers: $z^n$ = $r^n$ * e^(intheta)
• Roots: z^$\frac{1}{n}$ = r^$\frac{1}{n}$ * e^$\frac{i*(theta+2k*pi}{n}$) for k = 0, 1, ..., n-1

Special values:

• e^(i*pi) = -1 (Euler's identity)
• e^$\frac{i*pi}{2}$ = i
• e^(i2pi) = 1
• e^$\frac{i*pi}{3}$ = 1/2 + i*sqrt$\frac{3}{2}$

Conversion: a + ib = sqrt(a^{2+b}^2) * e^(i*arctan$\frac{b}{a}$) [with quadrant adjustment]