$${z = a + bi}$$
$${w = c + di}$$
$${|z| = \sqrt{a^2 + b^2} }$$
Complex conjugate:
$${\bar z = a - bi}$$
$${Re(z) = a}$$
$${Im(z) = b}$$
Euler's Formula:
$${e^{ix} = \cos(x) + i \sin(x) }$$
$${Re(z) = \frac{1}{2} (z + \bar z) }$$
$${Im(z) = \frac{1}{2i} (z - \bar z) }$$
$${|z|^2 = a^2 + b^2 = z \bar z }$$
$${|z| = \sqrt{z \bar z} }$$
$${\overline{z + w} = \bar z + \bar w }$$
$${\overline{z w} = \bar z \bar w }$$
$${z \bar w = \overline{\bar z w} }$$
$${\bar z w = \overline{z \bar w} }$$
$${|z \bar w| = |\bar z w| = |z| |w| }$$
$${Re(z \bar w) \leq |Re(z \bar w)| \leq |z \bar w| = |z| |w| }$$
Triangle Inequality:
$${|z + w| \leq |z| + |w| }$$
$${z = a + bi}$$
$${r = |z| = \sqrt{a^2 + b^2} }$$
$${a = r \cos(t)}$$
$${b = r \sin(t)}$$
$${t = arg(z) = \tan^{-1} \left( \frac{b}{a} \right) }$$
$${z = r e^{i \left(t + 2 \pi m \right)}}$$
$${z^n = (a + bi)^n = r^n e^{i\left(nt + 2 \pi m \right)} }$$
$${w^n = z }$$
$${w = \sqrt[n]{z} = \sqrt[n]{r} e^{i \left( \frac{t}{n} + 2 \pi \frac{m}{n} \right)}, m,n \in \mathbb{Z}^+ }$$
$${0 \leq m \lt n }$$
$${1 + w + w^2 + \cdots + w^{n-1} = 0 }$$
$${z^n = (a + bi)^n = \left( r e^{it} \right)^n = r^n e^{int} }$$
$${z^n = \left( r \cos t + r i \sin t \right)^n = r^n \left(\cos nt + i \sin nt \right) }$$
$${z = |z| e^{i arg(z)}}$$
$${\ln(z) = ln |z| + i \ arg(z) }$$
$${e^{i \pi} = -1 }$$
$${e^{i 2 \pi} = 1 }$$
$${e^{i \frac{\pi}{2} } = i }$$
$${e^{i \frac{3 \pi}{2} } = -i }$$
$${\sqrt{2} e^{i \frac{\pi}{4} } = 1 + i }$$
$${\sqrt{2} e^{-i \frac{\pi}{4} } = 1 - i }$$
$${e^{i \frac{\pi}{3} } = \frac{1}{2} + \frac{\sqrt{3}}{2} i }$$
$${i^2 = -1 }$$
$${i^3 = -i }$$
$${i^4 = 1 }$$