Complex Numbers

$\mathbb{C} = \{x + y \cdot i | x,y \in \mathbb{R}\}$ Set of complex numbers
$i^2 = -1$

Euler’s Formula

$e^{i \Theta} = cos(\Theta) + i \cdot sind(\Theta)$
Euler’s identity: $e^{i \cdot \pi} = -1$

Elementary Functions

Exponential function

$exp : \mathbb{R} \rightarrow \mathbb{R}$
We write: $exp(x) = e^x$

1. $$e^0 = 1$$
2. $$\forall x,y \in \mathbb{R}: e^{x+y} = e^x \cdot e^y$$
3. $$\forall x,y \in \mathbb{R}: e^x \neq 0 \land e^{-x}$$
4. $$\forall x \in \mathbb{R}: e^x > 0$$
5. exp grows monotonous
6. $exp: \mathbb{R} \rightarrow \]0,\infty\[$ is bijectiv