  The trace of the heat kernel is expanded in a basis of nonlocal curvature
invariants of $N$th order. The coefficients of this expansion (the nonlocal
form factors) are calculated to third order in the curvature inclusive. The
early-time and late-time asymptotic behaviours of the trace of the heat kernel
are presented with this accuracy. The late-time behaviour gives the criterion
of analyticity of the effective action in quantum field theory. The latter
point is exemplified by deriving the effective action in two dimensions.
