Euler's identity
$ e^{j \pi} + 1 = 0, \,\! $
Euler's formula
$ e^{jx} = \cos x + j \sin x \,\! $
$ \cos x = \mathrm{Re}\{e^{jx}\} ={e^{jx} + e^{-jx} \over 2} $
$ \sin x = \mathrm{Im}\{e^{jx}\} ={e^{jx} - e^{-jx} \over 2i}. $
$ \cos(x) = {e^{-jx} + e^{jx} \over 2} $
$ \sin(x) = {e^{-jx} - e^{jx} \over 2j} $
