Personal notes on a few complex functions.
In all of the sections below, \(z\) is a complex number, while \(x\) and \(y\) are real numbers.
Parts
\[ \mathrm{Re}(x+iy) = x \] \[ \mathrm{Im}(x+iy) = y \]
Modulus
\[ |x+iy| = \sqrt{x^2+y^2} \]
Conjugate
\[ \overline{x+iy} = x-iy \]
Exponential
\[ e^{x+iy} = e^x (\cos y + i \sin y) \]
Logarithm
\[ \mathrm{Log} \, z = \ln |z| + i \mathrm{Arg} \, z \]
Trigonometric
\[ \cos z = \frac{e^{iz} + e^{-iz}}{2} \] \[ \sin z = \frac{e^{iz} - e^{-iz}}{2i} \]