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Lambert W-function

Table of contents: Definition - Illustrations - Transcendental equations - Specific values - Symmetry - Analytic properties - Derivatives and integrals - Series expansions - Range - Bounds and inequalities

Definition

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

Illustrations

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Image: X-ray of W ⁣(z)W\!\left(z\right) on z∈[−3,3]+[−3,3]iz \in \left[-3, 3\right] + \left[-3, 3\right] i

Transcendental equations

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Wk ⁣(z)exp⁡ ⁣(Wk ⁣(z))=zW_{k}\!\left(z\right) \exp\!\left(W_{k}\!\left(z\right)\right) = z
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solutions⁡ w∈C[wew=z]={Wk ⁣(z):k∈Z  and⁡  (z≠0  or⁡  k=0)}\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\}
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solution*⁡ w∈[−1,∞)[wew=x]=W ⁣(x)\mathop{\operatorname{solution*}\,}\limits_{w \in \left[-1, \infty\right)} \left[w {e}^{w} = x\right] = W\!\left(x\right)
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solution*⁡ w∈(−∞,−1][wew=x]=W−1 ⁣(x)\mathop{\operatorname{solution*}\,}\limits_{w \in \left(-\infty, -1\right]} \left[w {e}^{w} = x\right] = W_{-1}\!\left(x\right)
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W ⁣(xex)=xW\!\left(x {e}^{x}\right) = x
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W−1 ⁣(xex)=xW_{-1}\!\left(x {e}^{x}\right) = x
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W ⁣(xlog⁡(x))=log⁡(x)W\!\left(x \log(x)\right) = \log(x)
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W−1 ⁣(xlog⁡(x))=log⁡(x)W_{-1}\!\left(x \log(x)\right) = \log(x)

Specific values

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W ⁣(0)=0W\!\left(0\right) = 0
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W ⁣(e)=1W\!\left(e\right) = 1
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W ⁣(−1e)=−1W\!\left(-\frac{1}{e}\right) = -1
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W−1 ⁣(−1e)=−1W_{-1}\!\left(-\frac{1}{e}\right) = -1
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W ⁣(1)∈[0.56714329040978387299996866221035554975381578718651±2.51⋅10−51]W\!\left(1\right) \in \left[0.56714329040978387299996866221035554975381578718651 \pm 2.51 \cdot 10^{-51}\right]
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W0′ ⁣(0)=1W'_{0}\!\left(0\right) = 1
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W0(r) ⁣(0)=(−r)r−1W^{(r)}_{0}\!\left(0\right) = {\left(-r\right)}^{r - 1}
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Wk ⁣(0)=−∞W_{k}\!\left(0\right) = -\infty
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W ⁣(−π2)=iπ2W\!\left(-\frac{\pi}{2}\right) = \frac{i \pi}{2}

Symmetry

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Wk ⁣(z‾)=W−k ⁣(z)‾W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}

Analytic properties

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W ⁣(z) is holomorphic on z∈C∖(−∞,−e−1]W\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, -{e}^{-1}\right]
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Wk ⁣(z) is holomorphic on z∈C∖(−∞,0]W_{k}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]
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poles⁡ z∈C∪{∞~}Wk ⁣(z)={}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} W_{k}\!\left(z\right) = \left\{\right\}
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BranchPoints⁡ ⁣(W0 ⁣(z),z,C∪{∞~})={−e−1,∞~}\operatorname{BranchPoints}\!\left(W_{0}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{-{e}^{-1}, {\tilde \infty}\right\}
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BranchPoints⁡ ⁣(Wk ⁣(z),z,C∪{∞~})={0,−e−1,∞~}\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, -{e}^{-1}, {\tilde \infty}\right\}
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BranchPoints⁡ ⁣(Wk ⁣(z),z,C∪{∞~})={0,∞~}\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, {\tilde \infty}\right\}
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BranchCuts⁡ ⁣(W0 ⁣(z),z,C)={(−∞,−e−1]}\operatorname{BranchCuts}\!\left(W_{0}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, -{e}^{-1}\right]\right\}
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BranchCuts⁡ ⁣(Wk ⁣(z),z,C)={(−∞,−e−1],[−e−1,0],(−∞,0]}\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\}
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BranchCuts⁡ ⁣(Wk ⁣(z),z,C)={(−∞,0]}\operatorname{BranchCuts}\!\left(W_{k}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}
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zeros⁡ z∈CW0 ⁣(z)={0}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{0}\!\left(z\right) = \left\{0\right\}
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zeros⁡ z∈CWk ⁣(z)={}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{k}\!\left(z\right) = \left\{\right\}

Derivatives and integrals

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Wk′ ⁣(z)=1(1+Wk ⁣(z))exp⁡ ⁣(Wk ⁣(z))W'_{k}\!\left(z\right) = \frac{1}{\left(1 + W_{k}\!\left(z\right)\right) \exp\!\left(W_{k}\!\left(z\right)\right)}
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Wk′ ⁣(z)=Wk ⁣(z)z(1+Wk ⁣(z))W'_{k}\!\left(z\right) = \frac{W_{k}\!\left(z\right)}{z \left(1 + W_{k}\!\left(z\right)\right)}

Series expansions

Taylor series

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W ⁣(z)=∑n=1∞(−n)n−1n!znW\!\left(z\right) = \sum_{n=1}^{\infty} \frac{{\left(-n\right)}^{n - 1}}{n !} {z}^{n}

Puiseux series

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Symbol: LambertWPuiseuxCoefficient — μk\mu_{k} — Coefficient in scaled Puiseux expansion of Lambert W-function
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Table of μk\mu_{k} for 0≤k≤150 \le k \le 15
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μk=k−1k+1(μk−22+αk−24)−αk2−μk−1k+1   where αk={2,k=0−1,k=1∑j=2k−1μjμk+1−j,otherwise\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}
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∣μk∣<2(45)k\left|\mu_{k}\right| < 2 {\left(\frac{4}{5}\right)}^{k}
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W ⁣(z)=∑n=0∞μnvn   where v=2(ez+1)W\!\left(z\right) = \sum_{n=0}^{\infty} \mu_{n} {v}^{n}\; \text{ where } v = \sqrt{2 \left(e z + 1\right)}
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Wk ⁣(z)=∑n=0∞μnvn   where v=−2(ez+1)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

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Wk ⁣(z)=L1⁡−L2⁡+∑n=0∞∑m=1∞(−1)nm![n+mn+1]σnτm   where L1⁡=log⁡(z)+2πik,  L2⁡=log⁡(L1⁡),  σ=1L1⁡,  τ=L2⁡L1⁡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}}
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∣Wk ⁣(z)−(L1⁡−L2⁡+∑n=0N−1∑m=1M−1(−1)nm![n+mn+1]σnτm)∣≤4∣τ∣(4∣σ∣)N+(4∣τ∣)M(1−4∣σ∣)(1−4∣τ∣)   where L1⁡=log⁡(z)+2πik,  L2⁡=log⁡(L1⁡),  σ=1L1⁡,  τ=L2⁡L1⁡\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

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{Wk ⁣(z):k∈Z  and⁡  z∈C  and⁡  (z≠0  or⁡  k=0)}=C\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}
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Wk1 ⁣(z1)≠Wk2 ⁣(z2)W_{{k}_{1}}\!\left({z}_{1}\right) \ne W_{{k}_{2}}\!\left({z}_{2}\right)

Image of the principal branch

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{W ⁣(x):x∈(−e−1,∞)}=(−1,∞)\left\{ W\!\left(x\right) : x \in \left(-{e}^{-1}, \infty\right) \right\} = \left(-1, \infty\right)
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{W ⁣(x):x∈{−e−1}}={−1}\left\{ W\!\left(x\right) : x \in \left\{-{e}^{-1}\right\} \right\} = \left\{-1\right\}
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{W ⁣(x):x∈(−∞,−e−1)}={−ycot⁡(y)+yi:y∈(0,π)}\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\}
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{W ⁣(z):z∈C∖R}={x+yi:y∈(−π,π)∖{0}  and⁡  x∈(−ycot⁡(y),∞)}\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

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{W−1 ⁣(z):z∈C∖{0}}=(−∞,−1]∪{x+yi:x∈R  and⁡  y∈R  and⁡  ((0<u<2  and⁡  t≤v)  or⁡  (1≤u≤2)  or⁡  (1<u<3  and⁡  t>v))   where t=xsinc⁡(y),  v=−cos⁡(y),  u=−yπ}\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\}
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{Wk ⁣(z):z∈C∖{0}}={x+yi:x∈R  and⁡  y∈R  and⁡  ((2k−2<u<2k  and⁡  t<v)  or⁡  (2k−1≤u≤2k)  or⁡  (2k−1<u<2k+1  and⁡  t≥v))   where t=xsinc⁡(y),  v=−cos⁡(y),  u=yπ}\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\}
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{W−k ⁣(z):z∈C∖{0}}={x+yi:x∈R  and⁡  y∈R  and⁡  ((2k−2<u<2k  and⁡  t≤v)  or⁡  (2k−1≤u≤2k)  or⁡  (2k−1<u<2k+1  and⁡  t>v))   where t=xsinc⁡(y),  v=−cos⁡(y),  u=−yπ}\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

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∣Im⁡ ⁣(W ⁣(z))∣<π\left|\operatorname{Im}\!\left(W\!\left(z\right)\right)\right| < \pi
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Im⁡ ⁣(W1 ⁣(z))∈(0,3π)\operatorname{Im}\!\left(W_{1}\!\left(z\right)\right) \in \left(0, 3 \pi\right)
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Im⁡ ⁣(W−1 ⁣(z))∈(−3π,0]\operatorname{Im}\!\left(W_{-1}\!\left(z\right)\right) \in \left(-3 \pi, 0\right]
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sgn⁡(k)Im⁡ ⁣(Wk ⁣(z))∈((2∣k∣−2)π,(2∣k∣+1)π)\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

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W0′ ⁣(x)≤1x+1W'_{0}\!\left(x\right) \le \frac{1}{x + 1}
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W0′ ⁣(x)<21+exW'_{0}\!\left(x\right) < \frac{2}{\sqrt{1 + e x}}
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∣W−1′ ⁣(x)∣<21+ex+2∣x∣\left|W'_{-1}\!\left(x\right)\right| < \frac{2}{\sqrt{1 + e x}} + \frac{2}{\left|x\right|}
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∣Wk′ ⁣(z)∣≤∣1.2z∣\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.2}{z}\right|
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∣Wk′ ⁣(z)∣≤∣1.5z∣\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.5}{z}\right|
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∣Wk′ ⁣(z)∣≤1∣z∣\left|W'_{k}\!\left(z\right)\right| \le \frac{1}{\left|z\right|}
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∣Wk′ ⁣(z)∣≤∣1z∣max⁡ ⁣(3,∣1.5∣ez+1∣∣)\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)
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∣W0′ ⁣(z)∣≤∣2.25t(1+t)∣   where t=∣ez+1∣\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|
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∣Wk′ ⁣(z)∣≤∣1∣z∣(1+14+∣z∣2)∣\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|
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∣Wk′ ⁣(z)∣≤∣1∣z∣(1+23321∣ez+1∣)∣\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