Generating a projection of Lambert’s Riemann surface

For each integer \(k\) there exists a branch of Lambert’s \(W\) denoted \(W_k\). The principal branch is \(W_0\) and the \((-1)\) branch is \(W_{-1}\). We will generate data for the three branches \(k=-1,\:0,\:1\) in sequence.

For the real projection of values of \(W\) we chose the sum of real and imaginary parts: \(z \mapsto \Re{(W(z))} + \Im{(W(z))}\). This was not the best choice, but it conforms to the usual choice we make to draw the Riemann surface of the logarithm.

For the domain we use a polar coordinate system centered at \(z= -\frac{ 1 }{ e }\) since that point is the edge of the negative \(x\)-axis branch cut, and the Riemann surface has a natural axial-rotational structure.

We sample values at approximately \(100\) radii ranging from \(0.01\) to \(4\) and approximately \(100\) angles sweeping everything but the negative \(x\)-semiaxis. The more samples, the longer the rendering will take, and this is already very slow even with lualatex. To generate the data, we execute with SageMath

#!/usr/bin/sage
# 
# Change shebang or pass to your `sage` executable according to
# your platform and installation.

radii    = [ x/100.0    for x in range(1, 400, 4)             ]
angles   = [ x/100000.0 for x in range(-314159, 314159, 5000) ]
branches = [ -1, 0, 1 ]

Lambert = [ [ (r*cos(t), r*sin(t), lambert_w(i, -1/e.n() + r * exp(I*t) ))
  for r in radii for t in angles ] for i in branches ]

LambertReal = [ [ (x, y, z.real() + z.imag())
  for (x,y,z) in Lambert[i] ] for i in branches ]

f = open(f"lambert.dat", "w")
_ = f.write("x y z\n")

for i in branches:
  for (a, b, c) in LambertReal[i]:
    _ = f.write(f"{a} {b} {c}\n")

and put the .dat file in the latex project folder. The rendering code in the tex file is given in the following page.

% !TeX TS-program = lualatex
%
% No command line options should be needed for lualatex with an
% up-to-date installation of TeXLive.  This file assumes that the
% generated `lambert.dat` file is in the same directory as this
% .tex file.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\begin{document}

\begin{tikzpicture}
    \begin{axis}
            [
                title={Part of the Riemann surface of $W$.},
                view={90}{10},
                width=7cm,
                height=10cm,
                axis line style={draw=none},
                ticks=none,
                colormap/violet,
                z buffer=sort,
            ]
        \addplot3[surf] table {lambert.dat};
    \end{axis}
\end{tikzpicture}

\end{document}