Fundamental Identities:

• z * z-bar = |z|^2 = $a^2$ + $b^2$
• z + z-bar = 2Re(z), z - z-bar = 2i*Im(z)
• z is real iff z = z-bar; purely imaginary iff z + z-bar = 0

Modulus Properties:

• |z1z2| = |z1||z2|, |z1/z2| = |z1|/|z2|, |$z^n$| = |z|^n
• |z-bar| = |z|, |z*z-bar| = |z|^2
• Triangle: ||z1|-|z2|| <= |z1+z2| <= |z1|+|z2|

Argument Properties:

• arg(z1*z2) = arg(z1) + arg(z2) mod 2pi
• arg$\frac{z1}{z2}$ = arg(z1) - arg(z2) mod 2pi
• arg(z-bar) = -arg(z), arg($z^n$) = n*arg(z)

Powers of i: $i^n$ = i^(n mod 4). Cycle: 1, i, -1, -i.

De Moivre: (cos t + i sin t)^n = cos(nt) + i sin(nt)

Euler: e^(i*theta) = cos theta + i sin theta

nth Roots: $z_k$ = r^$\frac{1}{n}$ * e^$\frac{i(theta+2k*pi}{n}$), k = 0,...,n-1

Cube Roots of Unity:

• 1 + w + $w^2$ = 0, $w^3$ = 1
• (1-w)(1-$w^2$) = 3
• a^{3+b}^3 = (a+b)(a+bw)(a+$bw^2$)

Rotation: z' = z0 + (z-z0)e^(ialpha) (about z0 by angle alpha)