Complex Numbers
\(\mathbb{C} = \{x + y \cdot i | x,y \in \mathbb{R}\}\) Set of complex numbers
\(i^2 = -1\)
\(e^{i \Theta} = cos(\Theta) + i \cdot sind(\Theta)\)
Euler’s identity: \(e^{i \cdot \pi} = -1\)
Elementary Functions
Exponential function
\(exp : \mathbb{R} \rightarrow \mathbb{R}\)
We write: \(exp(x) = e^x\)
-
\[e^0 = 1\]
-
\[\forall x,y \in \mathbb{R}: e^{x+y} = e^x \cdot e^y\]
-
\[\forall x,y \in \mathbb{R}: e^x \neq 0 \land e^{-x}\]
-
\[\forall x \in \mathbb{R}: e^x > 0\]
- exp grows monotonous
- \(exp: \mathbb{R} \rightarrow \]0,\infty\[\) is bijectiv