A complex number z=x+iy where x,y \in \mathbb{R}, z \in \mathbb{C}
x, Re(z) - real part of z
y, Im(z) - imaginary part of z
\mathbb{R} \subsetneq \mathbb{C}
i \in \mathbb{C} - \mathbb{R}
\bar{z} = x - iy
|z| = \sqrt{x^2 + y^2}
z = |z|(\cos{\alpha} + i\sin{\alpha})
where:
\cos{\alpha} = \frac{x}{|z|}
\sin{\alpha} = \frac{y}{|z|}
z = |z|e^{i\alpha}
z^k = |z|^ke^{ik\alpha}
therefore:
e^{iz} = \cos{z} + i\sin{z}
An n-th root of a complex number z (\sqrt[n]{z}) has n solutions:
\{w_0, w_1, \cdots, w_{n-1}\} these are
vertices of an n-regular polygon.
w_l = \sqrt[n]{|z|}e^{i\frac{\alpha + 2\pi l}{n}}
let f(x) = x^2 + x + 1
f has no real roots it has however, 2 complex ones:
\begin{matrix} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \\ x = \frac{-1 \pm \sqrt{-3}}{2} \\ x = \frac{-1 \pm \sqrt{3i^2}}{2} \\ x = -\frac{1}{2} \pm i\frac{\sqrt{3}}{2} \\ \end{matrix}
if a,b,c \in \mathbb{R} and z is a root of f then \bar{z} is a root as well
let z = |z|(\cos{\alpha} + i\sin{\alpha}), w = |w|(\cos{\phi} + i\sin{\phi}) then:
zw = |z||w|(\cos(\phi + \alpha) + i\sin(\phi + \alpha))
z^n = |z|^n(\cos(n\alpha) + i\sin(n\alpha))