Here is a visualization of several branches of the Lambert W function on the complex plane:

This function is defined implicitly as the inverse of the nonlinear transcendental equation

$W\left(z\right)e^{W\left(z\right)}=z$

An extremely comprehensive overview of this function, including details of evaluation and applications, is available here.

Evaluation of the function is accomplished using standard inversion algorithms, such as Newton’s method, but one needs a good starting point in order to reach all branches. To find this, first take a logarithm of both sides of the defining equation:

$W\left(z\right)=logz-logW\left(z\right)\approx logz+f\left(z\right)$

The Lambert function can thus be approximated by the natural logarithm, which explains the similarity of its appearance to that function. Putting the right-hand side into the defining equation and ignoring the additive term f in comparison to the logarithm on the resulting left-hand side,

$\begin{array}{c}(logz+f)z{e}^f\approx z{e}^{f}logz=z\\ f=-loglogz\end{array}$

which gives the approximation

$W\left(z\right)\approx logz-loglogz$

This form can be justified by putting it into a slightly rearranged defining equation,

$W\left(z\right)=z{e}^{-W\left(z\right)}\approx z{e}^{-logz+loglogz}=logz$

so that the second term of the approximation allows one to recover the first term.

All that remains now is to specify the branches of the logarithms. This can be done by keeping the second outer logarithm on the principal branch and letting the index of the Lambert function set the branch of the other functions. That is,

$W_n\left(z\right)\approx logz+2\pi in-log(logz+2\pi in)$

Starting from this complex point allows one to determine the value of the Lambert function for all branches, with the exception of the principal branch W₀ in the vicinity of the origin. Since the function is zero there, starting from that value reaches most of the region around the origin, apart from some instability around the negative real axis.

Here is a comparison of the approximation (in red) to the final evaluated result:

While the Lambert function is similar to the natural logarithm, the branches of its imaginary part are not equally spaced as for the latter. This can be seen by visualizing multiple branches of the imaginary part at the same time:

The Lambert function has a complex structure with respect to argument similar to that of the natural logarithm, but reaches its asymptotic coloring noticeably faster than that function.

Uploaded 2020.11.24 — Updated 2020.12.25 analyticphysics.com