ca_atan_logarithm, ca_atan_direct, ca_atan

The logarithm version writes $\operatorname{atan} x = \frac{i}{2} (\log(1 - ix) - \log(1 + ix))$ (in a form that respects the branch cuts), the direct version returns the generator of $\mathbb{Q}(\operatorname{atan} x)$, possibly after transforming the argument. Simple algebraic arguments such as $1$ or $\sqrt{3}$ give rational multiples of $\pi$. ca_atan uses the logarithm form unless CA_OPT_TRIG_FORM is CA_TRIG_DIRECT.

For special values: $\operatorname{atan}(\pm i) = \pm i \infty$, $\operatorname{atan}(c \infty) = \operatorname{csgn}(c) \pi / 2$, and $\operatorname{atan}(\tilde \infty)$ is Undefined.

@misc{flint-doc-ca-ca-atan-logarithm,
  author       = {{The FLINT team}},
  title        = {{ca\_atan\_logarithm, ca\_atan\_direct, ca\_atan (ca.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/ca/ca_atan_logarithm.html},
  note         = {Generated 2026-11-02}
}