We consider a family of conformally symplectic maps, which are characterized by the property that they transform a symplectic form into a multiple of itself. We assume that the conformal factor depends on a parameter, such that we recover the symplectic case when the parameter goes to zero. We study the perturbative expansions and the domains of analyticity in the symplectic limit of the parameterization of the quasi--periodic orbits with Diophantine frequency. Our main result is to prove that the tori are analytic in a domain in the complex parameter plane, obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The proof is based on developing a theorem in an "a-posteriori" format, that is used to validate (under certain conditions) the formal asymptotic expanions. The rigorous results match very well recent numerical explorations that, in turn, suggest new conjectures. Joint work with R. Calleja and R. de la Llave