# Exponential function

## Definitions

Symbol: Exp ${e}^{z}$ Exponential function
Symbol: ConstE $e$ The constant e (2.718...)

## Illustrations

Image: Plot of ${e}^{x}$ on $x \in \left[-4, 4\right]$
Image: X-ray of ${e}^{z}$ on $z \in \left[-5, 5\right] + \left[-5, 5\right] i$

## Domain and range

### Numbers

$z \in \left\{0\right\} \;\implies\; {e}^{z} \in \left\{1\right\}$
$z \in \mathbb{R} \;\implies\; {e}^{z} \in \left(0, \infty\right)$
$z \in \mathbb{C} \;\implies\; {e}^{z} \in \mathbb{C} \setminus \left\{0\right\}$

### Infinities

$z \in \left\{\infty\right\} \;\implies\; {e}^{z} \in \left\{\infty\right\}$
$z \in \left\{-\infty\right\} \;\implies\; {e}^{z} \in \left\{0\right\}$

### Formal power series

$\left(z \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z = 0\right) \;\implies\; \left({e}^{z} \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] {e}^{z} = 1\right)$
$z \in \mathbb{R}[[x]] \;\implies\; \left({e}^{z} \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] {e}^{z} \ne 0\right)$
$z \in \mathbb{C}[[x]] \;\implies\; \left({e}^{z} \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] {e}^{z} \ne 0\right)$

### Matrices

$A \in \operatorname{M}_{n \times n}\!\left(\mathbb{R}\right) \;\implies\; {e}^{A} \in \operatorname{GL}_{n}\!\left(\mathbb{R}\right)$
$A \in \operatorname{M}_{n \times n}\!\left(\mathbb{C}\right) \;\implies\; {e}^{A} \in \operatorname{GL}_{n}\!\left(\mathbb{C}\right)$

## Particular values

${e}^{0} = 1$
${e}^{1} = e$
${e}^{\pi i} = -1$
${e}^{\pi i / 2} = i$
$e \notin \mathbb{Q}$
$e \notin \overline{\mathbb{Q}}$
${e}^{\alpha} \notin \mathbb{Q} \;\text{ for all } \alpha \in \mathbb{Q} \setminus \left\{0\right\}$
${e}^{\alpha} \notin \overline{\mathbb{Q}} \;\text{ for all } \alpha \in \overline{\mathbb{Q}} \setminus \left\{0\right\}$

## Functional equations and connection formulas

${e}^{a + b} = {e}^{a} {e}^{b}$
${\left({e}^{z}\right)}^{n} = {e}^{n z}$
${e}^{-z} = \frac{1}{{e}^{z}}$
${e}^{a + b i} = {e}^{a} \left(\cos(b) + \sin(b) i\right)$
${e}^{z + n \pi i} = {\left(-1\right)}^{n} {e}^{z}$
${e}^{z + 2 n \pi i} = {e}^{z}$
${e}^{z} = \cosh(z) + \sinh(z)$
${e}^{i z} = \cos(z) + i \sin(z)$
${e}^{\log(z)} = z$
$\operatorname{Im}(z) \in \left(-\pi, \pi\right] \;\implies\; \left(\log\!\left({e}^{z}\right) = z\right)$

## Analytic properties

${e}^{z} \text{ is holomorphic on } z \in \mathbb{C}$
$\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} {e}^{z} = \left\{\right\}$
$\operatorname{EssentialSingularities}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left({e}^{z}, z, \mathbb{C}\right) = \left\{\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} {e}^{z} = \left\{\right\}$

## Complex parts

$\left|{e}^{z}\right| = {e}^{\operatorname{Re}(z)}$
$\operatorname{sgn}\!\left({e}^{z}\right) = {e}^{\operatorname{Im}(z) i}$
$\operatorname{Re}\!\left({e}^{z}\right) = {e}^{\operatorname{Re}(z)} \cos\!\left(\operatorname{Im}(z)\right)$
$\operatorname{Im}\!\left({e}^{z}\right) = {e}^{\operatorname{Re}(z)} \sin\!\left(\operatorname{Im}(z)\right)$
$\arg\!\left({e}^{z}\right) = \operatorname{Im}(z)$
$\exp\!\left(\overline{z}\right) = \overline{{e}^{z}}$

## Taylor series

${e}^{z} = \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}$
${e}^{c + z} = {e}^{c} \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}$

## Derivatives and integrals

$\int_{a}^{b} {e}^{z} \, dz = {e}^{b} - {e}^{a}$
$\frac{d}{d z}\, {e}^{z} = {e}^{z}$
$\frac{d^{n}}{{d z}^{n}} {e}^{z} = {e}^{z}$

## Approximations

$\left|{e}^{z} - \sum_{k=0}^{N - 1} \frac{{z}^{k}}{k !}\right| \le \frac{{\left|z\right|}^{N}}{N ! \left(1 - \frac{\left|z\right|}{N}\right)}$

## Bounds and inequalities

$\left|{e}^{z}\right| \le {e}^{\left|z\right|}$
$\left|{e}^{x + a} - {e}^{x}\right| \le {e}^{x} \left({e}^{\left|a\right|} - 1\right)$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC