Difference between revisions of "Exponential"
From specialfunctionswiki
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| − | + | 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!},$$ |
| − | + | where $e$ is the [[E | base of the natural logarithm]]. It can be characterized as the unique solution to the initial value problem | |
$$\left\{ \begin{array}{ll} | $$\left\{ \begin{array}{ll} | ||
y'=y \\ | y'=y \\ | ||
| Line 7: | Line 7: | ||
\end{array} \right.$$ | \end{array} \right.$$ | ||
| − | + | <div align="center"> | |
| + | <gallery> | ||
| + | File:Exp.png|Graph of $\mathrm{arccos}$ on $\mathbb{R}$. | ||
| + | File:Complex exp.jpg|[[Domain coloring]] of [[analytic continuation]] of $\exp$. | ||
| + | </gallery> | ||
| + | </div> | ||
| − | [[ | + | [[500px]] |
Revision as of 14:04, 1 November 2014
The exponential function $\exp \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\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 $\exp$.
