Exponential

Euler constant e

$$\exp^{x}$$

is called the natural exponential function

$$e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots e= 2.718 281 828...$$

$\exp(ix)$

$$\exp^{ix} = cos(x) + i sin(x)$$ $$\exp^{ix} = \frac{\exp^{ix} + \exp^{-ix}}{2} + i\cdot\frac{\exp^{ix} - \exp^{-ix}}{2i}$$ $$\exp^{ix} = \frac{\exp^{ix} + \exp^{-ix}}{2} + i\cdot\frac{\exp^{ix} - \exp^{-ix}}{2i}$$ $$\exp^{ix} = \frac{\exp^{ix} + \exp^{-ix}}{2} + \frac{\exp^{ix} - \exp^{-ix}}{2}$$ $$\exp^{ix} = \frac{\exp^{ix} + \exp^{-ix} + \exp^{ix} - \exp^{-ix}}{2}$$ $$\exp^{ix} = \frac{\exp^{ix} + \exp^{ix} + \exp^{-ix} - \exp^{-ix}}{2}$$ $$\exp^{ix} = \frac{\exp^{ix} + \exp^{ix}}{2}$$ $$\exp^{ix} = \frac{2\exp^{ix}}{2}$$ $$\exp^{ix} = \exp^{ix}$$

$\exp(-ix)$

$$\exp^{-ix} = cos(x) - i sin(x)$$ $$\exp^{-ix} = \frac{\exp^{ix} + \exp^{-ix}}{2} - i\cdot\frac{\exp^{ix} - \exp^{-ix}}{2i}$$ $$\exp^{-ix} = \frac{\exp^{ix} + \exp^{-ix}}{2} - i\cdot\frac{\exp^{ix} - \exp^{-ix}}{2i}$$ $$\exp^{-ix} = \frac{\exp^{ix} + \exp^{-ix}}{2} - \frac{\exp^{ix} - \exp^{-ix}}{2}$$ $$\exp^{-ix} = \frac{\exp^{ix} + \exp^{-ix} - (\exp^{ix} - \exp^{-ix})}{2}$$ $$\exp^{-ix} = \frac{\exp^{ix} + \exp^{-ix} - \exp^{ix} + \exp^{-ix}}{2}$$ $$\exp^{-ix} = \frac{\exp^{ix} - \exp^{ix} + \exp^{-ix} + \exp^{-ix}}{2}$$ $$\exp^{-ix} = \frac{\exp^{-ix} + \exp^{-ix}}{2}$$ $$\exp^{-ix} = \frac{2\exp^{-ix}}{2}$$ $$\exp^{-ix} = \exp^{-ix}$$