Numerical Analysis of Lambert Function

Question

For a mathematical project I want to analyse the following function: $R_{\infty} = 1 + \frac{W(\epsilon e^{-\epsilon})}{\epsilon}$, in which the $W$ stands for the Lambert function. I have no experience with this function and after some research on internet I'm still stuck. I need to analyse this equation, thus I need to find a relation between $R$ and $\epsilon$. How can I do this? I tried mathematica for a numerical analysis but I don't get it to work.

Any help would be appreciated!

Answer 1

For "small" values of $x$, you could use Padé approximants built at $x=0$ and get $$R(x) = 1 + \frac{W(x \, e^{-x})}{x}\approx \frac{2+6x+\frac{8 }{3}x^2 }{1+4x+\frac{10 }{3}x^2}$$ which is quite good up to $x=2$ (for this value of $x$, the exact value of the function is $\approx 1.10886 $ while the approximation gives $\frac {74 } {67 }\approx 1.10448$).

Answer 2

I am using $x$ instead of $\epsilon$. With $R(x) = 1 + \frac{W(x e^{-x})}{x},$ Maple gives the two asymptotic expressions:

$$R(x) \sim 2-2x+4x^2-\tfrac{28}{3}x^3+24x^4+O(x^5), \quad x\rightarrow 0$$

$$R(x) \sim 1+e^{-x}-xe^{-2x}+\tfrac{3}{2}x^2e^{-3x}-\tfrac{8}{3}x^3e^{-4x}+O(x^4e^{-5x}), \quad x \rightarrow \infty$$

Here the used Maple commands

> f := x ->1 + LambertW(x*exp(-x))/x;
                                                              LambertW(x exp(-x))
                                                f := x -> 1 + -------------------
                                                                       x
> series(f(x),x=0);
                                                          2         3       4      5
                                             2 - 2 x + 4 x  - 28/3 x  + 24 x  + O(x )
> series(f(x), x=infinity);
                                                                2           3           4
                                          1         x      3/2 x           x           x
                                    1 + ------ - ------- + ------- - 8/3 ------- + O(-------)
                                        exp(x)         2         3             4           5
                                                 exp(x)    exp(x)        exp(x)      exp(x)

If you want to compute / approximate the $W$ function in the intermediate range, have a look at Having Fun with Lambert W(x) Function and https://github.com/DarkoVeberic/LambertW