  Conformally symplectic systems include mechanical systems with a friction
proportional to the velocity. Geometrically, these systems transform a
symplectic form into a multiple of itself making the systems dissipative or
expanding. In the present work we consider the limit of small dissipation. The
example we study is a family of conformally symplectic standard maps of the
cylinder for which the conformal factor, $b(\varepsilon)$, is a function of a
small complex parameter, $\varepsilon$.
  We assume that for $\varepsilon=0$ the map preserves the symplectic form and
the dependence on $\varepsilon$ is cubic, i.e., $b(\varepsilon) = 1 -
\varepsilon^3$. We compute perturbative expansions formally in $\varepsilon$
and use them to estimate the shape of the domains of analyticity of invariant
circles as functions of $\varepsilon$. We also give evidence that the functions
might belong to a Gevrey class at $\varepsilon = 0$. We also perform numerical
continuation of the solutions as they pass through the boundary of the domain
to illustrate that the monodromy of the solutions is trivial. The numerical
computations we perform support conjectures on the shape of the domains of
analyticity.
