12/10/2007, 04:11 AM
For more information about the Lambert W function, which this thread seems to be a great deal about, Wolfram's function website has a wealth of information about W and about the branches of W (you can even download PDFs of each section). To answer Ivars' question earlier in this thread, the method probably used to find the power series expansions of the Lambert W function is probably to apply general analytic continuation techniques, or to solve what coefficients satisfy the following differential equation:
)W'(x) = W(x))
as this uniquely defines the Lambert W function along with initial conditions.
The general series expansion of the Lambert W function for all branches is:
 = w <br />
+ \frac{w}{(1+w)x_0}(x-x_0)<br />
- \frac{2w^2+w^3}{2(1+w)^3x_0^2}(x-x_0)^2<br />
+ \frac{9w^3+8w^4+2w^5}{6(1+w)^5x_0^3}(x-x_0)^3<br />
+ \cdots)
which can be obtained by repeated application of the derivative formula:
 = \frac{W(x)}{x(1+W(x))})
from which the differential equation came. So all you need is the starting value )
at the starting point 
and you can form a series expansion of W anywhere. These are just the formulas I found on Wolfram's function website.
Andrew Robbins
The general series expansion of the Lambert W function for all branches is:
Andrew Robbins