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Lambert's W-Function


The inverse of the function

\end{displaymath} (1)

also called the Omega Function. The function is implemented as the Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) function ProductLog[z]. $W(1)$ is called the Omega Constant and can be considered a sort of ``Golden Ratio'' of exponentials since
\mathop{\rm exp}\nolimits [-W(1)]=W(1),
\end{displaymath} (2)

\ln\left[{1\over W(1)}\right]=W(1).
\end{displaymath} (3)

Lambert's $W$-Function has the series expansion

W(x)=\sum_{n=1}^\infty {(-1)^{n-1} n^{n-2}\over(n-1)!} x^n =...
...xtstyle{54\over 5}}x^6+{\textstyle{16807\over 720}}x^7+\ldots.
\end{displaymath} (4)

The Lagrange Inversion Theorem gives the equivalent series expansion
W_0(z)=\sum_{n=1}^\infty {(-n)^{n-1}\over n!}z^n,
\end{displaymath} (5)

where $n!$ is a Factorial. However, this series oscillates between ever larger Positive and Negative values for Real $z\mathrel{\hbox{\hbox to 0pt{%
\lower.5ex\hbox{$\sim$}\hss}\raise.4ex\hbox{$>$}}} 0.4$, and so cannot be used for practical numerical computation. An asymptotic Formula which yields reasonably accurate results for $z\mathrel{\hbox{\hbox to 0pt{%
\lower.5ex\hbox{$\sim$}\hss}\raise.4ex\hbox{$>$}}}3$ is

$\displaystyle W(z)$ $\textstyle =$ $\displaystyle \mathop{\rm Ln}\nolimits z-\ln\mathop{\rm Ln}\nolimits z+\sum_{k=...
..._{km}(\ln\mathop{\rm Ln}\nolimits z)^{m+1}(\mathop{\rm Ln}\nolimits z)^{-k-m-1}$  
  $\textstyle =$ $\displaystyle L_1-L_2+{L_2\over L_1}+{L_2(-2+L_2)\over 2{L_1}^2}+{L_2(6-9L_2+2{L_2}^2\over 6{L_1}^2}$  
  $\textstyle \phantom{=}$ $\displaystyle +{L_2(-12+36L_2-22{L_2}^2+3{L_2}^3)\over 12{L_1}^4}$  
  $\textstyle \phantom{=}$ $\displaystyle +{L_2(60-300L_2+350{L_2}^2-125{L_2}^3+12{L_2}^4)\over 60{L_1}^5}+{\mathcal O}\left[{\left({L_2\over L_1}\right)^6}\right],$ (6)

$\displaystyle L_1$ $\textstyle =$ $\displaystyle \mathop{\rm Ln}\nolimits z$ (7)
$\displaystyle L_2$ $\textstyle =$ $\displaystyle \ln\mathop{\rm Ln}\nolimits z$ (8)

(Corless et al.), correcting a typographical error in de Bruijn (1961). Another expansion due to Gosper is the Double Sum

W(x)=a+\sum_{n=0}^\infty \left\{{\sum_{k=0}^n {S_1(n,k)\over...
...ight\} \left[{1-{\ln\left({x\over a}\right)\over a}}\right]^n,
\end{displaymath} (9)

where $S_1$ is a nonnegative Stirling Number of the First Kind and $a$ is a first approximation which can be used to select between branches. Lambert's $W$-function is two-valued for $-1/e\leq x<0$. For $W(x)\geq -1$, the function is denoted $W_0(x)$ or simply $W(x)$, and this is called the principal branch. For $W(x)\leq -1$, the function is denoted $W_{-1}(x)$. The Derivative of $W$ is
W'(x)={1\over [1+W(x)]\mathop{\rm exp}\nolimits [W(x)]}={W(x)\over x[1+W(x)]}
\end{displaymath} (10)

for $x\not=0$. For the principal branch when $z>0$,
\ln W(z)=\ln z-W(z).
\end{displaymath} (11)

See also Iterated Exponential Constants, Omega Constant


de Bruijn, N. G. Asymptotic Methods in Analysis. Amsterdam, Netherlands: North-Holland, pp. 27-28, 1961.

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© 1996-9 Eric W. Weisstein