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In this paper, based on lectures by the authors at the May 2015 workshop {\itResurgence, Physics and Numbers}, at the Centro di Ricerca Matematica Ennio DeGiorgio of the Scuola Normale Superiore in Pisa, we explain the origin ofresurgent trans-series in the Mathieu equation spectral problem, using uniformWKB and all-orders (exact) WKB. Exact quantization conditions naturally arise,and their expansion in the weak coupling regime produces resurgent trans-seriesexpressions which exhibit precise relations between different instantonsectors. Indeed, the perturbative expansion encodes all orders of themulti-instanton expansion, an explicit realization of the general concept of"resurgence". We also discuss the transition from weak to strong coupling, anexplicit realization of "instanton condensation".

WKB and Resurgence in the Mathieu Equation