# Natural logarithm

## Definitions

Symbol: Log $\log\!\left(z\right)$ Natural logarithm

## Illustrations

Image: X-ray of $\log\!\left(z\right)$ on $z \in \left[-3, 3\right] + \left[-3, 3\right] i$

## Particular values

$\log\!\left(1\right) = 0$
$\log\!\left(e\right) = 1$
Table of $\log\!\left(n\right)$ to 50 digits for $1 \le n \le 50$
$\log\!\left(i\right) = \frac{\pi i}{2}$
$\log\!\left(-1\right) = \pi i$

## Functional equations and connection formulas

$\exp\!\left(\log\!\left(z\right)\right) = z$
$\log\!\left({e}^{z}\right) = z$
$\log\!\left(z\right) = \log\!\left(\left|z\right|\right) + \arg\!\left(z\right) i$
$\log\!\left(c z\right) = \log\!\left(c\right) + \log\!\left(z\right)$

## Analytic properties

$\operatorname{HolomorphicDomain}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, 0\right]$
$\operatorname{Poles}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{EssentialSingularities}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchPoints}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}, 0\right\}$
$\operatorname{BranchCuts}\!\left(\log\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}$
$\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) + \pi i$
$\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) - \pi i$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log\!\left(z\right) = \left\{1\right\}$

## Complex parts

$\log\!\left(\overline{z}\right) = \overline{\log\!\left(z\right)}$
$\operatorname{Re}\!\left(\log\!\left(z\right)\right) = \log\!\left(\left|z\right|\right)$
$\operatorname{Im}\!\left(\log\!\left(z\right)\right) = \arg\!\left(z\right)$
$\left|\log\!\left(z\right)\right| = \sqrt{{\left(\log\!\left(\left|z\right|\right)\right)}^{2} + {\left(\arg\!\left(z\right)\right)}^{2}}$

## Bounds and inequalities

$\log\!\left(x\right) \le x - 1$
$\left|\log\!\left(z\right)\right| \le \left|\log\!\left(\left|z\right|\right)\right| + \pi$
$\left|\log\!\left(x + a\right) - \log\!\left(x\right)\right| \le \log\!\left(1 + \frac{\left|a\right|}{x - \left|a\right|}\right)$

## Integral representations

$\log\!\left(z\right) = \int_{1}^{z} \frac{1}{t} \, dt$

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