Asymptotics of subtracted singularities for generating functions with small singularities

We derive uniform asymptotic results for generating functions with small singularities. A typical example is the generating function of permutations with no cycles of length smaller than m, 1⩽m⩽c ln n. These asymptotic expressions are the same as the ones we would obtain applying singularity analysis (if we could apply it when m is not fixed). In addition to permutations with no cycles of length smaller than m, the method can be applied, for instance, to several generating functions for polynomials over finite fields. We exemplify this with squarefree polynomials over finite fields having no irreducible factor of degree smaller than m, and with polynomials over finite fields with smallest irreducible factor of degree equal to m, 1⩽m⩽c ln n. Our framework is based on the method of subtracted singularities that has its genesis in the method of Darboux (J. Math. Pures Appl. 4 (1878)).