Quantified versions of Ingham’s Tauberian theorem

We present several quantified versions of Ingham’s Tauberian theorem in
which the rate of decay is determined by the behaviour of a certain boundary
function near its singularities. The proofs of these results are modified
versions of Ingham’s own proof and, in particular, involve no estimates of
contour integrals. The general results are then applied in the setting of C_0-
semigroups, giving both new proofs of previously known results and, in one
important case, a sharper result than was previously available.