Does the Lambert W function tend to infinity?

Question

Does $W(x)\to\infty$ as the real number $x\to+\infty$?

I find the equation (4.19) in paper https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf. It shows $$W(x)=\log x-\log\log x+\cdots$$.

Assuming $x\in\mathbb{R}^+$, $x\to+\infty$, is the following equation $$\frac{1}{1+W(x)}\to0$$ OK? Or it convergesto a certain value?

In fact $W(x) > \log x - \log\log x$ for $x > e$. See, e.g., this answer of mine. — Antonio Vargas, Dec 12 '13 at 03:10

score 2 · Answer 1 · answered Dec 12 '13 at 03:05

2

$W(\infty)=x\iff x\,e^x=\infty$ . Obviously, if x were finite, then so were $\infty$, which is absurd.

answered Dec 12 '13 at 03:05

Lucian

I like that "absurd" is a mathematically correct thing to say. – Simply Beautiful Art May 19 '16 at 00:35

score 0 · Answer 2 · answered Dec 12 '13 at 02:54

0

Yes, it goes to infinity. If you have mathematica (or Wolfram Alpha), the Lambert function is available as ProductLog[]

answered Dec 12 '13 at 02:54

Igor Rivin

2 Answers2