acb_lambertw

With flags equal to ACB_LAMBERTW_LEFT, the function computes $W_{\mathrm{left}|k}(z)$, which is $W_k(z)$ in the upper half plane and $W_{k+1}(z)$ in the lower half plane, connected continuously to the left of the branch points; the branch cut on $(-\infty, 0)$ is rotated counterclockwise to $(0, +\infty)$. (For $k = -1$ and $k = 0$ there is also a branch cut on $(-1/e, 0)$, continuous from below instead of from above to keep counterclockwise continuity.)

With ACB_LAMBERTW_MIDDLE, it computes $W_{\mathrm{middle}}(z)$, which is $W_{-1}(z)$ in the upper half plane and $W_1(z)$ in the lower half plane, connected continuously through $(-1/e, 0)$, with branch cuts on $(-\infty, -1/e)$ and $(0, +\infty)$. It extends the real analytic function $W_{-1}(x)$ on $(-1/e, 0)$ to a complex analytic function, whereas the standard $W_{-1}(z)$ has a branch cut on that segment.

@misc{flint-doc-acb-acb-lambertw,
  author       = {{The FLINT team}},
  title        = {{acb\_lambertw (acb.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/acb/acb_lambertw.html},
  note         = {Generated 2026-11-02}
}