It is usually defined as the solution to W(x)e^W(x) = x. It is analytic everywhere except at x = -1/e. Its series expansion is the (seemingly nice) one W(z) = Σ (-n)^n-1 / n! z^n however, this is divergent except for values of z quite close to 0.

A way cool function, W(z), defined:

f(W) = We^W ⇒ W(f(z)) = z

As a sort of golden ratio of exponentials, the omega constant:

ω = W(1) = 0.56714...

Satisfies:

e^-ω = ω

Which is equivilent to a lot of fun things otherwise unsolvable in exponentials.

The Lambert W Function can be used to express the analytic result of the infinite power tower:

x^xx... = -W(-ln x)/ln x

Its derivative is:

W(x)/[x(1 + W(x))]

And its integral cannot be expressed in terms of itself and elementary functions.

(From Eric Weisstein's MathWorld, http://mathworld.wolfram.com/LambertsW-Function.html)