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# Lambert W-function

## Definition

Symbol: LambertW $W_{k}\!\left(z\right)$ Lambert W-function

## Illustrations

Image: X-ray of $W_{0}\!\left(z\right)$ on $z \in \left[-3, 3\right] + \left[-3, 3\right] i$

## Transcendental equations

$W_{k}\!\left(z\right) \exp\!\left(W_{k}\!\left(z\right)\right) = z$
$\mathop{\operatorname{solutions}\,}\limits_{w \in \mathbb{C}} \left[w {e}^{w} = z\right] = \left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\}$
$\mathop{\operatorname{solution}\,}\limits_{w \in \left[-1, \infty\right)} \left[w {e}^{w} = x\right] = W_{0}\!\left(x\right)$
$\mathop{\operatorname{solution}\,}\limits_{w \in \left(-\infty, -1\right]} \left[w {e}^{w} = x\right] = W_{-1}\!\left(x\right)$
$W_{0}\!\left(x {e}^{x}\right) = x$
$W_{-1}\!\left(x {e}^{x}\right) = x$
$W_{0}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)$
$W_{-1}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)$

## Specific values

$W_{0}\!\left(0\right) = 0$
$W_{0}\!\left(e\right) = 1$
$W_{0}\!\left(-\frac{1}{e}\right) = -1$
$W_{-1}\!\left(-\frac{1}{e}\right) = -1$
$W_{0}\!\left(1\right) \in \left[0.56714329040978387299996866221035554975381578718651 \pm 2.51 \cdot 10^{-51}\right]$
$W'_{0}\!\left(0\right) = 1$
$W^{(r)}_{0}\!\left(0\right) = {\left(-r\right)}^{r - 1}$
$W_{k}\!\left(0\right) = -\infty$
$W_{0}\!\left(-\frac{\pi}{2}\right) = \frac{i \pi}{2}$

## Symmetry

$W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}$

## Analytic properties

$\operatorname{HolomorphicDomain}\!\left(W_{0}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, -{e}^{-1}\right]$
$\operatorname{HolomorphicDomain}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, 0\right]$
$\operatorname{Poles}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchPoints}\!\left(W_{0}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{-{e}^{-1}, {\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, -{e}^{-1}, {\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, {\tilde \infty}\right\}$
$\operatorname{BranchCuts}\!\left(W_{0}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, -{e}^{-1}\right]\right\}$
$\operatorname{BranchCuts}\!\left(W_{k}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, -{e}^{-1}\right], \left[-{e}^{-1}, 0\right], \left(-\infty, 0\right]\right\}$
$\operatorname{BranchCuts}\!\left(W_{k}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{0}\!\left(z\right) = \left\{0\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{k}\!\left(z\right) = \left\{\right\}$

## Derivatives and integrals

$W'_{k}\!\left(z\right) = \frac{1}{\left(1 + W_{k}\!\left(z\right)\right) \exp\!\left(W_{k}\!\left(z\right)\right)}$
$W'_{k}\!\left(z\right) = \frac{W_{k}\!\left(z\right)}{z \left(1 + W_{k}\!\left(z\right)\right)}$

## Series expansions

### Taylor series

$W_{0}\!\left(z\right) = \sum_{n=1}^{\infty} \frac{{\left(-n\right)}^{n - 1}}{n !} {z}^{n}$

### Puiseux series

Symbol: LambertWPuiseuxCoefficient ${\mu}_{k}$ Coefficient in scaled Puiseux expansion of Lambert W-function
Table of ${\mu}_{k}$ for $0 \le k \le 15$
${\mu}_{k} = \frac{k - 1}{k + 1} \left(\frac{{\mu}_{k - 2}}{2} + \frac{{\alpha}_{k - 2}}{4}\right) - \frac{{\alpha}_{k}}{2} - \frac{{\mu}_{k - 1}}{k + 1}\; \text{ where } {\alpha}_{0} = 2,\,{\alpha}_{1} = -1,\,{\alpha}_{k} = \sum_{j=2}^{k - 1} {\mu}_{j} {\mu}_{k + 1 - j}$
$\left|{\mu}_{k}\right| \lt 2 {\left(\frac{4}{5}\right)}^{k}$
$W_{0}\!\left(z\right) = \sum_{n=0}^{\infty} {\mu}_{n} {v}^{n}\; \text{ where } v = \sqrt{2 \left(e z + 1\right)}$
$W_{k}\!\left(z\right) = \sum_{n=0}^{\infty} {\mu}_{n} {v}^{n}\; \text{ where } v = -\sqrt{2 \left(e z + 1\right)}$

### Logarithmic expansion

$W_{k}\!\left(z\right) = {L}_{1} - {L}_{2} + \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}$
$\left|W_{k}\!\left(z\right) - \left({L}_{1} - {L}_{2} + \sum_{n=0}^{N - 1} \sum_{m=1}^{M - 1} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\right)\right| \le \frac{4 \left|\tau\right| {\left(4 \left|\sigma\right|\right)}^{N} + {\left(4 \left|\tau\right|\right)}^{M}}{\left(1 - 4 \left|\sigma\right|\right) \left(1 - 4 \left|\tau\right|\right)}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}$

## Range

### Tiling of the plane

$\left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\} = \mathbb{C}$
$W_{{k}_{1}}\!\left({z}_{1}\right) \ne W_{{k}_{2}}\!\left({z}_{2}\right)$

### Image of the principal branch

$\left\{ W_{0}\!\left(x\right) : x \in \left(-{e}^{-1}, \infty\right) \right\} = \left(-1, \infty\right)$
$\left\{ W_{0}\!\left(x\right) : x \in \left\{-{e}^{-1}\right\} \right\} = \left\{-1\right\}$
$\left\{ W_{0}\!\left(x\right) : x \in \left(-\infty, -{e}^{-1}\right) \right\} = \left\{ -y \cot\!\left(y\right) + y i : y \in \left(0, \pi\right) \right\}$
$\left\{ W_{0}\!\left(z\right) : z \in \mathbb{C} \setminus \mathbb{R} \right\} = \left\{ x + y i : y \in \left(-\pi, \pi\right) \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \left(-y \cot\!\left(y\right), \infty\right) \right\}$

### Image of the non-principal branches

$\left\{ W_{-1}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left(-\infty, -1\right] \cup \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(0 \lt u \lt 2 \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(1 \le u \le 2\right) \,\mathbin{\operatorname{or}}\, \left(1 \lt u \lt 3 \,\mathbin{\operatorname{and}}\, t \gt v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}$
$\left\{ W_{k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 \lt u \lt 2 k \,\mathbin{\operatorname{and}}\, t \lt v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \ge v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = \frac{y}{\pi} \right\}$
$\left\{ W_{-k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 \lt u \lt 2 k \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \gt v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}$

## Bounds and inequalities

### Complex parts

$\left|\operatorname{Im}\!\left(W_{0}\!\left(z\right)\right)\right| \lt \pi$
$\operatorname{Im}\!\left(W_{1}\!\left(z\right)\right) \in \left(0, 3 \pi\right)$
$\operatorname{Im}\!\left(W_{-1}\!\left(z\right)\right) \in \left(-3 \pi, 0\right]$
$\operatorname{Im}\!\left(W_{k}\!\left(z\right)\right) \in \left(\operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| - 2\right) \pi, \operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| + 1\right) \pi\right)$

### Derivative bounds

$W'_{0}\!\left(x\right) \le \frac{1}{x + 1}$
$W'_{0}\!\left(x\right) \lt \frac{2}{\sqrt{1 + e x}}$
$\left|W'_{-1}\!\left(x\right)\right| \lt \frac{2}{\sqrt{1 + e x}} + \frac{2}{\left|x\right|}$
$\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.2}{z}\right|$
$\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.5}{z}\right|$
$\left|W'_{k}\!\left(z\right)\right| \le \frac{1}{\left|z\right|}$
$\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{z}\right| \max\!\left(3, \left|\frac{1.5}{\sqrt{\left|e z + 1\right|}}\right|\right)$
$\left|W'_{0}\!\left(z\right)\right| \le \left|\frac{2.25}{\sqrt{t \left(1 + t\right)}}\right|\; \text{ where } t = \left|e z + 1\right|$
$\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{1}{4 + {\left|z\right|}^{2}}\right)\right|$
$\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{23}{32} \frac{1}{\sqrt{\left|e z + 1\right|}}\right)\right|$

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