acb_log_sin_pi

Equivalently, $S(z) = \int_{1/2}^z \pi \cot(\pi t)\, dt$ with the path through the upper half plane if $0 < \arg z \le \pi$ and through the lower half plane otherwise, or

with the negative sign if $0 < \arg z \le \pi$ (the union of both cases is computed when $\arg z$ is not determined). After subtracting $n$, $0 \le \operatorname{Re} z < 1$, and $S(z) = \log \sin(\pi z)$ is used if the imaginary part is small; otherwise $S(z) = i \pi (z - 1/2) + \log((1 + e^{-2 \pi i z})/2)$ in the lower half plane and the conjugate expression in the upper half plane, which avoid overflow. The function is evaluated at the midpoint and the propagated error is bounded with $S'(z)$, so that the result changes continuously when $z$ is not real and $n$ is not determined.

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