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# Lambert W-function

## Definition

Symbol: LambertW $W\!\left(z\right)$ Lambert W-function

## Illustrations

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

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

## Specific values

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

## Symmetry

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

## Analytic properties

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

## 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

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

### Puiseux series

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

### Logarithmic expansion

$W_{k}\!\left(z\right) = \operatorname{L_1} - \operatorname{L_2} + \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\; \text{ where } \operatorname{L_1} = \log(z) + 2 \pi i k,\;\operatorname{L_2} = \log(\operatorname{L_1}),\;\sigma = \frac{1}{\operatorname{L_1}},\;\tau = \frac{\operatorname{L_2}}{\operatorname{L_1}}$
$\left|W_{k}\!\left(z\right) - \left(\operatorname{L_1} - \operatorname{L_2} + \sum_{n=0}^{N - 1} \sum_{m=1}^{M - 1} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\right)\right| \le \frac{4 \left|\tau\right| {\left(4 \left|\sigma\right|\right)}^{N} + {\left(4 \left|\tau\right|\right)}^{M}}{\left(1 - 4 \left|\sigma\right|\right) \left(1 - 4 \left|\tau\right|\right)}\; \text{ where } \operatorname{L_1} = \log(z) + 2 \pi i k,\;\operatorname{L_2} = \log(\operatorname{L_1}),\;\sigma = \frac{1}{\operatorname{L_1}},\;\tau = \frac{\operatorname{L_2}}{\operatorname{L_1}}$

## 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

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

### Image of the non-principal branches

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

## Bounds and inequalities

### Complex parts

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

### Derivative bounds

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

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

2021-03-15 19:12:00.328586 UTC