  We discuss two theorems in analytic number theory and combinatory analysis
that have seen increased use in recent years. A corollary to a Tauberian
theorem of Ingham allows one to quickly prove asymptotic formulas for
arithmetic sequences, so long as the corresponding generating function exhibits
exponential growth of a certain form near its radius of convergence. Two common
methods for proving the required analytic behavior are modular transformations
and Euler-Maclaurin summation.
  However, these results are sometimes stated without certain technical
conditions that are necessary for the complex analytic techniques that appear
in Ingham's proof. We carefully examine the precise statements and proofs of
these results, and find that in practice, the technical conditions are
satisfied for those cases appearing in recent applications. We also generalize
the classical approach of Euler-Maclaurin summation in order to prove
asymptotic expansions for series with complex values, simple poles, or
multi-dimensional summation indices.
