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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 Wk ⁣(z)W_{k}\!\left(z\right) Lambert W-function

Illustrations

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Image: X-ray of W0 ⁣(z)W_{0}\!\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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solutionswC[wew=z]={Wk ⁣(z):kZand(z0ork=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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solutionw[1,)[wew=x]=W0 ⁣(x)\mathop{\operatorname{solution}\,}\limits_{w \in \left[-1, \infty\right)} \left[w {e}^{w} = x\right] = W_{0}\!\left(x\right)
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solutionw(,1][wew=x]=W1 ⁣(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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W0 ⁣(xex)=xW_{0}\!\left(x {e}^{x}\right) = x
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W1 ⁣(xex)=xW_{-1}\!\left(x {e}^{x}\right) = x
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W0 ⁣(xlog ⁣(x))=log ⁣(x)W_{0}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)
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W1 ⁣(xlog ⁣(x))=log ⁣(x)W_{-1}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)

Specific values

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W0 ⁣(0)=0W_{0}\!\left(0\right) = 0
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W0 ⁣(e)=1W_{0}\!\left(e\right) = 1
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W0 ⁣(1e)=1W_{0}\!\left(-\frac{1}{e}\right) = -1
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W1 ⁣(1e)=1W_{-1}\!\left(-\frac{1}{e}\right) = -1
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W0 ⁣(1)[0.56714329040978387299996866221035554975381578718651±2.511051]W_{0}\!\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)r1W^{(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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W0 ⁣(π2)=iπ2W_{0}\!\left(-\frac{\pi}{2}\right) = \frac{i \pi}{2}

Symmetry

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

Analytic properties

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HolomorphicDomain ⁣(W0 ⁣(z),z,C{~})=C(,e1]\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]
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HolomorphicDomain ⁣(Wk ⁣(z),z,C{~})=C(,0]\operatorname{HolomorphicDomain}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, 0\right]
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Poles ⁣(Wk ⁣(z),z,C{~})={}\operatorname{Poles}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchPoints ⁣(W0 ⁣(z),z,C{~})={e1,~}\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,e1,~}\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)={(,e1]}\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)={(,e1],[e1,0],(,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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zeroszCW0 ⁣(z)={0}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{0}\!\left(z\right) = \left\{0\right\}
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zeroszCWk ⁣(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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W0 ⁣(z)=n=1(n)n1n!znW_{0}\!\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 0k150 \le k \le 15
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μk=k1k+1(μk22+αk24)αk2μk1k+1   where α0=2,α1=1,αk=j=2k1μjμk+1j{\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}
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μk<2(45)k\left|{\mu}_{k}\right| \lt 2 {\left(\frac{4}{5}\right)}^{k}
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W0 ⁣(z)=n=0μnvn   where v=2(ez+1)W_{0}\!\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=0μnvn   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)=L1L2+n=0m=1(1)nm![n+mn+1]σnτm   where L1=log ⁣(z)+2πik,L2=log ⁣(L1),σ=1L1,τ=L2L1W_{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}}
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Wk ⁣(z)(L1L2+n=0N1m=1M1(1)nm![n+mn+1]σnτm)4τ(4σ)N+(4τ)M(14σ)(14τ)   where L1=log ⁣(z)+2πik,L2=log ⁣(L1),σ=1L1,τ=L2L1\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

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{Wk ⁣(z):kZandzCand(z0ork=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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{W0 ⁣(x):x(e1,)}=(1,)\left\{ W_{0}\!\left(x\right) : x \in \left(-{e}^{-1}, \infty\right) \right\} = \left(-1, \infty\right)
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{W0 ⁣(x):x{e1}}={1}\left\{ W_{0}\!\left(x\right) : x \in \left\{-{e}^{-1}\right\} \right\} = \left\{-1\right\}
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{W0 ⁣(x):x(,e1)}={ycot ⁣(y)+yi:y(0,π)}\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\}
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{W0 ⁣(z):zCR}={x+yi:y(π,π){0}andx(ycot ⁣(y),)}\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

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{W1 ⁣(z):zC{0}}=(,1]{x+yi:xRandyRand((0<u<2andtv)or(1u2)or(1<u<3andt>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 \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\}
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{Wk ⁣(z):zC{0}}={x+yi:xRandyRand((2k2<u<2kandt<v)or(2k1u2k)or(2k1<u<2k+1andtv))   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 \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\}
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{Wk ⁣(z):zC{0}}={x+yi:xRandyRand((2k2<u<2kandtv)or(2k1u2k)or(2k1<u<2k+1andt>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 \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

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Im ⁣(W0 ⁣(z))<π\left|\operatorname{Im}\!\left(W_{0}\!\left(z\right)\right)\right| \lt \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 ⁣(W1 ⁣(z))(3π,0]\operatorname{Im}\!\left(W_{-1}\!\left(z\right)\right) \in \left(-3 \pi, 0\right]
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Im ⁣(Wk ⁣(z))(sgn ⁣(k)(2k2)π,sgn ⁣(k)(2k+1)π)\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

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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) \lt \frac{2}{\sqrt{1 + e x}}
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W1 ⁣(x)<21+ex+2x\left|W'_{-1}\!\left(x\right)\right| \lt \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)1z\left|W'_{k}\!\left(z\right)\right| \le \frac{1}{\left|z\right|}
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Wk ⁣(z)1zmax ⁣(3,1.5ez+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)1z(1+14+z2)\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)1z(1+23321ez+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.

2019-07-15 23:42:41.550119 UTC