What uses does the product log function have?

Question

I've been looking into complex functions and how to plot them in programing languages like Python and JavaScript. I still am wondering how to do stuff with complex functions like in my previous question. I was looking around and found an interesting function called the product log function. It's the inverse of $\space xe^x \space$ also written $W(x)$. While trying to find more information on this function I found that it's not the only function that is an inverse to $\space xe^x \space$ there is a whole list of them, when I say list I mean countable infinity. They are labeled $W_n(x)$ where $n$ is an integer. This makes sense since the natural log $\ln(x)$ has a similar property where $\ln(x)+2\pi n$ is also an inverse of $e^x$. You can see the product logs structure quite nicely in it's Riemann surface. It looks like a spiral digging down to one point.

I've seen the natural log be used in several places in science and engineering, but haven't seen the product logs used in many areas or fields. This function looks very nice. Where are the product logs used in applied fields of study?

Answer 1

To make the story short, any equation which can write or rewrite $$a +b x +c \log(d x+e)=0$$ has explicit solutions in terms of the standard Lambert function. $$x=-\frac{e}{d}+\frac{c }{b}\,W(k) \qquad \text{where} \qquad k=\frac b {cd}\,\exp\Big[\frac{b e-a d}{c d}\Big]$$

If, on the search bar of the site, you just type Lambert, you will find $3569$ entries.

Answer 2

A thorough view of Lambert's $W$ function is given by the paper that gave it it's name: Corless, Gonnet, Hare, Jeffrey, Knuth "On the Lambert $W$ Function", Advances in Computational Mathematics 5, pages 329–359 (1996). A draft copy can be found here.

It turns out to have a surprising array of uses. Simpler ones are to solve equations in which the indeterminate appears both as a term and in an exponent.