What is Lambert W (e^\xi-1=x\xi) Function?

[CRGreathouse]

I ran across a function and wondered if it's named. It reminds me of Lambert's W. I quote from my source (van de Lune and Wattel 1969):

where $\xi$ is the positive root of $e^\xi-1=x\xi$

Here $x$ is a constant for our purposes.

The function is essentially logarithmic in the limit, as is clear from its definition.

 

[HallsofIvy]

"Where $\zeta$ is the positive root of $e^{/zeta}- 1= x/zeta$" doesn't define anything! Surely there was a definition before the "where"!

 

[Matthew Rodman]

Don't know what it's called, but I gave a solution of it elsewhere[sosmath.com] (sorry 'bout the plug for a rival website ).

 

[CRGreathouse]

"Where $\zeta$ is the positive root of $e^{/zeta}- 1= x/zeta$" doesn't define anything! Surely there was a definition before the "where"!

Huh? First of all, that's not what I wrote; in addition to the typographic difference (I used xi, you used zeta) you divide where I multiply. But it does define a function* -- or rather a family of functions, one for each value of x. The function is $f_x:\mathbb{R}^+\to\mathbb{R}$, defined by

$$f_x(z)=\xi \Leftrightarrow e^\xi-1=x\xi$$

Of course this definition doesn't show that the function is single-valued on the reals, nor that it's defined for all positive reals, or the like... that's one reason I'd like to find out if there are 'known properties' of this function.

* And it's a good thing, too, since the thing that comes before the "where" was another function defined using $\xi$.

 

[CRGreathouse]

Don't know what it's called, but I gave a solution of it elsewhere[sosmath.com]

OK, so you suggest

$$-\xi=\frac1x+W\left(\frac{-\exp(-1/x)}{x}\right)$$

Any thought on which branch to take? The W should be defined on both branches for x > 0.

 

[CRGreathouse]

where $\xi$ is the positive root of $e^\xi-1=x\xi$

Here $x$ is a constant for our purposes.

The function is essentially logarithmic in the limit, as is clear from its definition.

Actually I'm interested in the behavior of the function around small values, maybe 3 to 20. In this neighborhood it's not really logarithmic -- any thoughts on how best to characterize it?