Perturbation theory is seldom enough to completely describe physical systems. The often asymptotic nature of the perturbative series is a warning for the need of a nonperturbative completion. In the theory of resurgence, physical quantities are described by transseries: formal series that potentially include perturbative and nonperturbative effects. To connect with the original physical quantities the formal transseries must be resummed into actual functions.
This resummation procedure, from formal to finite, is illustrated for a matrix model with a quartic potential. The free energy has a large N expansion in the 't Hooft limit that has been extended to a resurgent transseries in recent years. We set up the resummation of the transseries and compare against exact results for the matrix integral at finite, small values of the rank N. We explore the complex plane for the 't Hooft coupling, deal with Stokes phenomena, and use the transseries to define values of free energies for noninteger rank.
We finish by showing how these techniques can be used in principle in more complicated theories like topological strings.
