Rapid Revision Bullets

$i^2$ = -1, powers cycle: 1, i, -1, -i (period 4)
z*z-bar = |z|^2 -- THE most important identity
z + z-bar = 2Re(z), z - z-bar = 2i*Im(z)
|z1z2| = |z1||z2|; arg(z1*z2) = arg(z1)+arg(z2)
Euler: e^(itheta) = cos(theta) + isin(theta)
De Moivre: (cis theta)^n = cis(n*theta)
nth roots: n equally spaced points on circle, sum = 0
Cube roots: 1+w+ $w^2$ = 0, $w^3$ = 1
1+w = - $w^2$ , 1+ $w^2$ = -w (use immediately)
|z-z0| = r is a circle; |z-z1| = |z-z2| is perpendicular bisector
Max|z| = |center| + radius; Min|z| = ||center| - radius|
Triangle inequality: ||z1|-|z2|| <= |z1+z2| <= |z1|+|z2|
arg(z-bar) = -arg(z); arg in (-pi, pi]
Complex roots of real polynomials come in conjugate pairs
sqrt(-a)*sqrt(-b) = -sqrt(ab), NOT sqrt(ab)
On unit circle: 1/z = z-bar
z+1/z = 2cos(theta) when z = e^(i*theta)
Rotation by alpha about origin: multiply by e^(i*alpha)
z = x+iy and separate parts for z + z-bar equations
Equilateral triangle: $z1^{2+z2}^{2+z3}^2$ = z1z2+z2z3+z3z1