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?
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$\begingroup$ If it's numerical methods you're after, there's some handy results in TOMS743 : dl.acm.org/citation.cfm?id=203084 $\endgroup$– JCKJun 1, 2019 at 22:40
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$\begingroup$ Now I found a new reference: Edwards, S: Extension of Algebraic Solutions Using The Lambert W Function. 2019 arxiv.org/abs/1902.08910 $\endgroup$– IV_Jul 6, 2019 at 19:41
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2 Answers
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This paper on Lambert W functions by Corless, Knuth and others should help.
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$\begingroup$ 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? $\endgroup$– IV_Dec 25, 2017 at 15:18
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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 $x^ae^{-bx^c}<e^{-B}$ 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 $x>B/W_0(Be^{-a})$.
answered Jun 1, 2019 at 15:23