Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e))

[andydude]

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:

\( x(1+W(x))W'(x) = W(x) \)

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:

\( W(x) = w
+ \frac{w}{(1+w)x_0}(x-x_0)
- \frac{2w^2+w^3}{2(1+w)^3x_0^2}(x-x_0)^2
+ \frac{9w^3+8w^4+2w^5}{6(1+w)^5x_0^3}(x-x_0)^3
+ \cdots \)

which can be obtained by repeated application of the derivative formula:

\( W'(x) = \frac{W(x)}{x(1+W(x))} \)

from which the differential equation came. So all you need is the starting value \( w = W_k(x_0) \) at the starting point \( x_0 \) and you can form a series expansion of W anywhere. These are just the formulas I found on Wolfram's function website.

Andrew Robbins