  The Brjuno function arises naturally in the study of one--dimensional
analytic small divisors problems. It belongs to $\hbox{BMO}({\Bbb T}^{1})$ and
it is stable under H\"older perturbations. It is related to the size of Siegel
disks by various rigorous and conjectural results.
  In this work we show how to extend the Brjuno function to a holomorphic
function on ${\Bbb H}/{\Bbb Z}$, the complex Brjuno function. This has an
explicit expression in terms of a series of transformed dilogarithms under the
action of the modular group.
  The extension is obtained using a complex analogue of the continued fraction
expansion of a real number. Since our method is based on the use of
hyperfunctions it applies to less regular functions than the Brjuno function
and it is quite general.
  We prove that the harmonic conjugate of the Brjuno function is bounded. Its
trace on ${\Bbb R}/{\Bbb Z}$ is continuous at all irrational points and has a
jump of $\pi/q$ at each rational point $p/q\in {\Bbb Q}$.
