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We consider curvature-induced resonances in a planar two-dimensional Dirichlet tube of a width $ d $. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as $ d\to 0+ $, provided the curvature of the strip axis satisfies certain analyticity and decay requirements., Comment: LaTeX2.09, 30 pages, 3 figures, to appear in Helv. Phys. Acta