The asymptotic behaviour of the partition function is one of the central questions of statistical mechanics. In our work we consider this problem when the external potential is complex valued and for a particular statistical-mechanical model, a β-ensemble. We prove a full 1/N expansion of the logarithm of the partition function, the so-called free energy. Our method can be regarded as an infinite dimensional version of the method of steepest descent for contour integrals. This is joint work with A. Guionnet and K. Kozlowski.