# What are the most general methods for solving equations in closed form with Lambert W?

What are the most general methods for solving equations with help of Lambert W function or with a generalization of Lambert W function in closed form?

I gave a method in MSE here.

Which algorithms are used e.g. by the computer algebra systems?

• If it's numerical methods you're after, there's some handy results in TOMS743 : dl.acm.org/citation.cfm?id=203084 – JCK Jun 1 at 22:40
• Now I found a new reference: Edwards, S: Extension of Algebraic Solutions Using The Lambert W Function. 2019 arxiv.org/abs/1902.08910 – IV_ Jul 6 at 19:41

## 2 Answers

This paper on Lambert W functions by Corless, Knuth and others should help.

https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf

• A collection of example equations is not a complete general solution theory. In particular, the relation to equations of elementary functions in general is missing. I assume the most general approach could be Rosenlicht's "On the explicit solvability of certain transcendental equations" for solving in differential fields. But are there further very general methods known? – IV_ Dec 25 '17 at 15:18

I have found the following (very easy) result to be absolutely essential in estimating the number of terms to take in multiprecision computations:

Let as usual $$W_0$$ and $$W_{-1}$$ be the two branches of the Lambert function.

1) Let $$a\in\mathbb R$$, $$b>0$$, $$c>0$$, and $$B>0$$. The solution to the inequality $$x^ae^{-bx^c} with $$x\ge(a/bc)^{1/c}$$ is given by: $$\begin{cases} x>((-a/(bc))W_{-1}(-(bc/a)e^{-(c/a)B}))^{1/c}&\text{ if a>0}\\ x>(B/b)^{1/c}&\text{ if a=0}\\ x>((-a/(bc))W_0(-(bc/a)e^{-(c/a)B}))^{1/c}&\text{ if a<0.}\end{cases}$$ In particular, in all cases, as $$B\to\infty$$ we have $$x>((1/b)(B+(a/c)\log(B/b)))^{1/c}+o(1).$$

2) If $$a\in\mathbb R$$ and $$B>0$$, the solution to the inequality $$x\log(x)-ax>B$$ with $$x>0$$ is $$x>B/W_0(Be^{-a})$$.