Lambert's W-Function

The inverse of the function

$$f(W)=We^W,$$ (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

$$\mathop{\rm exp}\nolimits [-W(1)]=W(1),$$ (2)

giving

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

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

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

The Lagrange Inversion Theorem gives the equivalent series expansion

$$W_0(z)=\sum_{n=1}^\infty {(-n)^{n-1}\over n!}z^n,$$ (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

$$W(z) = \mathop{\rm Ln}\nolimits z-\ln\mathop{\rm Ln}\nolimits z+\sum_{k=... ..._{km}(\ln\mathop{\rm Ln}\nolimits z)^{m+1}(\mathop{\rm Ln}\nolimits z)^{-k-m-1}$$

$$= L_1-L_2+{L_2\over L_1}+{L_2(-2+L_2)\over 2{L_1}^2}+{L_2(6-9L_2+2{L_2}^2\over 6{L_1}^2}$$

$$\phantom{=} +{L_2(-12+36L_2-22{L_2}^2+3{L_2}^3)\over 12{L_1}^4}$$

$$\phantom{=} +{L_2(60-300L_2+350{L_2}^2-125{L_2}^3+12{L_2}^4)\over 60{L_1}^5}+{\mathcal O}\left[{\left({L_2\over L_1}\right)^6}\right],$$ (6)

where

$$L_1 = \mathop{\rm Ln}\nolimits z$$ (7)

$$L_2 = \ln\mathop{\rm Ln}\nolimits z$$ (8)

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

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

$$W'(x)={1\over [1+W(x)]\mathop{\rm exp}\nolimits [W(x)]}={W(x)\over x[1+W(x)]}$$ (10)

for $x\not=0$. For the principal branch when $z>0$,

$$\ln W(z)=\ln z-W(z).$$ (11)

See also Iterated Exponential Constants, Omega Constant

References

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