  We apply exact WKB methods to the study of the partition function of pure N=2
epsilon_i-deformed gauge theory in four dimensions in the context of the 2d/4d
correspondence. We study the partition function at leading order in
epsilon_2/epsilon_1 (i.e. at large central charge) and in an expansion in
epsilon_1. We find corrections of the form ~ exp[- SW periods / epsilon_1] to
this expansion. We attribute these to the exchange of the order of summation
over gauge instanton number and over powers of epsilon_1 when passing from the
Nekrasov form of the partition function to the topological string theory
inspired form. We conjecture that such corrections should be computable from a
worldsheet perspective on the partition function. Our results follow upon the
determination of the Stokes graphs associated to the Mathieu equation with
complex parameters and the application of exact WKB techniques to compute the
Mathieu characteristic exponent.
