# Natural logarithm

## Definitions

Symbol: Log $\log(z)$ Natural logarithm

## Illustrations

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

## Particular values

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

## Functional equations and connection formulas

${e}^{\log(z)} = z$
$\log\!\left({e}^{z}\right) = z$
$\log\!\left({e}^{z}\right) = z - 2 \pi i \left\lceil \frac{\operatorname{Im}(z)}{2 \pi} - \frac{1}{2} \right\rceil$
$\log(z) = \log\!\left(\left|z\right|\right) + \arg(z) i$
$\log\!\left(c z\right) = \log(c) + \log(z)$

## Analytic properties

$\log(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]$
$\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \log(z) = \left\{\right\}$
$\operatorname{EssentialSingularities}\!\left(\log(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchPoints}\!\left(\log(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}, 0\right\}$
$\operatorname{BranchCuts}\!\left(\log(z), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}$
$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) + \pi i$
$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) - \pi i$
$\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i$
$\mathop{\text{Continuation}}\limits_{\displaystyle{t: 0 \rightsquigarrow \theta}} \, \log\!\left(R {e}^{i t}\right) = \log(R) + \theta i$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log(z) = \left\{1\right\}$

## Complex parts

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

## Bounds and inequalities

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

## Integral representations

$\log(z) = \int_{1}^{z} \frac{1}{t} \, dt$
$\int \frac{1}{z} \, dz = \log(z) + \mathcal{C}$
$\int \frac{1}{z} \, dz = \log\!\left(-z\right) + \mathcal{C}$
$\int \frac{1}{x} \, dx = \log\!\left(\left|x\right|\right) + \mathcal{C}$

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