  We prove that the solutions of a cohomological equation of complex dimension
one and in the analytic category have a monogenic dependence on the parameter,
and we investigate the question of their quasianalyticity. This equation is the
standard linearized conjugacy equation for germs of holomorphic maps in a
neighborhood of a fixed point. The parameter is the eigenvalue $q$ of the
linear part. This problem has been first investigated by Arnol'd and Herman.
Herman raised the question whether the solutions of the cohomological equation
had a quasianalytic dependence on the parameter. Indeed they are analytic
outside $\S^1$ which is a natural boundary but the solutions are still defined
at points of $\S^1$ which lie ``far enough from resonances''. We adapt to our
case Herman's construction of an increasing sequence of compacts which avoid
resonances and prove that the solutions belong to the associated space of
monogenic functions. The solutions admit asymptotic expansions at the points of
$\S^1$ which satisfy some arithmetical condition, and Carleman's Theorem allows
us to answer negatively to the question of quasianalyticity at these points.
But resonances (roots of unity) lead to asymptotic expansions, for which
quasianalyticity is obtained as a particular case of \'Ecalle's theory of
resurgent functions. At constant-type points one can still recover the
solutions from their asymptotic expansions and obtain a special kind of
quasianalyticity. Our results are obtained by reducing the problem to the study
of a fundamental solution (which is the ``quantum logarithm''). We deduce as a
corollary of our work the proof of a conjecture of Gammel on the monogenic and
quasianalytic properties of a certain number-theoretical Borel-Wolff-Denjoy
series.
