  Asymptotic expansions were first introduced by Henri Poincare in 1886. This
paper describes their application to the semi-classical evaluation of
amplitudes in quantum field theory with boundaries. By using zeta-function
regularization, the conformal anomaly for a massless spin-${1\over 2}$ field in
flat Euclidean backgrounds with boundary is obtained on imposing locally
supersymmetric boundary conditions. The quantization program for gauge fields
and gravitation in the presence of boundaries is then introduced by focusing on
conformal anomalies for higher-spin fields. The conditions under which the
covariant Schwinger-DeWitt and the non-covariant, mode-by-mode analysis of
quantum amplitudes agree are described.
