# Better approximation for Lambert W Function near zero

I've looking for a polynomial approximation for Lambert W Function around zero.

I am aiming at the range of $$0\leq x\leq e$$, and if possible then even $$-e\leq x\leq e$$.

The asymptotic expansion (when limited to a finite number of iterations of course), yields a polynomial approximation for the range of $$-1/e\leq x\leq1/e$$.

I've found the following polynomial approximation for the range of $$0\leq x\leq e$$ in this answer:

$$W_0(z)\approx\ln(1+z)\frac{1+\frac{123}{40}z+\frac{21}{10}z^2}{1+\frac{143}{40}z+\frac{713}{240}z^2}$$

I've tested it to my satisfaction, but I would like ask a few questions here:

1. Can it be easily extended to the range of $$-e\leq x\leq e$$?
2. The author of that answers explains something about Padé approximant, and says that one might do a better approximation (which he/she did not bother to find due to the question in context). Would anyone be able to shed light on how to compute a better polynomial approximation?
3. Is there perhaps a different approach to my goal, unrelated of the answer above?

Thank you!

• These are not polynomial approximations. They are rational approximations to $W(x)/\ln(1+x)$. Jun 9, 2020 at 18:30

## 1 Answer

The minimax rational approximation of $$W(x)/\ln(1+x)$$ on $$[0,e]$$ with numerator and denominator of degree $$2$$ is, according to Maple, $$\frac{0.0396202320 + 0.1961951280 x + 0.1702729841 x^2}{0.0396188863 + 0.2161222712 x + 0.2405866129 x^2}$$ which has maximum absolute error approximately $$0.00003396612388$$ on that interval.

You're not going to be able to get an approximation on $$[-e, e]$$ because $$W(x)$$ has a branch point at $$-1/e$$ and all its branches are complex for $$x < -1/e$$. You could look for approximations on $$[-1/e, e]$$. Thus the minimax rational approximation of $$W(x)/\ln(1+x)$$ on $$[-1/e, e]$$ with numerator and denominator of degree $$2$$ is

$$\frac{0.0663274708 + 0.2304903265 x + 0.1716240158 x^2}{0.0663270883 + 0.2636065188 x + 0.2426002763 x^2}$$ with maximum absolute error approximately $$0.00004705730323$$ on that interval.

You can get better accuracy with higher-degree approximations.

• Thank you! Your last (second) approximation is for $x\in[-1/e,e]$? Jun 9, 2020 at 18:45
• Yes, that's what I said. Jun 9, 2020 at 18:47
• Awesome (and very useful) answer, thank you very much!!! Jun 9, 2020 at 19:33
• I would like to ask one more thing: since I am already handling the range $[-1/e,1/e]$ accurately, I technically need to handle only the range $[1/e,e]$. Will Maple generate a more accurate 2nd-degree approximation if I narrow the input range in your first example? Jun 9, 2020 at 19:44
• Yes, on a smaller interval you should get a better approximation. In this case, for degrees $2$ and $2$ Maple is telling me "error curve fails to oscillate sufficiently; try different degrees". With numerator of degree $1$ and denominator $3$, it gives me $${\frac { 0.276972170484786+ 0.332660424982433\,x}{ 0.277527230334148 + 0.466289676939177\,x+ 0.00150785702317642\,{x}^{2}- 0.000244712398927012\,{x}^{3}}}$$ with maximum error $6.281026638 \times 10^{-6}$. Jun 9, 2020 at 23:53