  We introduce a new numerical method to compute resonances induced by
localized defects in crystals. This method solves an integral equation in the
defect region to compute analytic continuations of resolvents. Such an approach
enables one to express the resonance in terms of a "resonance source", a
function that is strictly localized within the defect region. The kernel of the
integral equation, to be applied on such a source term, is the Green function
of the perfect crystal, which we show can be computed efficiently by a complex
deformation of the Brillouin zone, named Brillouin Complex Deformation (BCD),
thereby extending to reciprocal space the concept of complex coordinate
transformations.
