# Exponential function

## Definitions

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

## Illustrations

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

## Particular values

${e}^{0} = 1$
${e}^{1} = e$
${e}^{\pi i} = -1$
${e}^{\pi i / 2} = i$

## Functional equations and connection formulas

${e}^{a + b} = {e}^{a} {e}^{b}$
${\left({e}^{z}\right)}^{n} = {e}^{n z}$
${e}^{-z} = \frac{1}{{e}^{z}}$
${e}^{a + b i} = {e}^{a} \left(\cos\!\left(b\right) + \sin\!\left(b\right) i\right)$
${e}^{z + n \pi i} = {\left(-1\right)}^{n} {e}^{z}$
${e}^{z + 2 n \pi i} = {e}^{z}$
${e}^{z} = \cosh\!\left(z\right) + \sinh\!\left(z\right)$
${e}^{i z} = \cos\!\left(z\right) + i \sin\!\left(z\right)$

## Analytic properties

$\operatorname{HolomorphicDomain}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C}$
$\operatorname{Poles}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{EssentialSingularities}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left({e}^{z}, z, \mathbb{C}\right) = \left\{\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} {e}^{z} = \left\{\right\}$

## Complex parts

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

## 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.

2019-06-18 07:49:59.356594 UTC