Exponential

The exponential function $\exp \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\exp(z) = e^z = \sum_{k=0}^{\infty} \dfrac{x^k}{k!},$$ where $e$ is the base of the natural logarithm. It can be characterized as the unique solution to the initial value problem $$\left\{ \begin{array}{ll} y'=y \\ y(0)=1. \end{array} \right.$$

• Exp.png

  Graph of $\mathrm{arccos}$ on $\mathbb{R}$.

• Complex exp.jpg

  Domain coloring of analytic continuation of $\exp$.