  One method to reconstruct the scalar field potential of inflation is a
perturbative approach, where the values of the potential and its derivatives
are calculated as an expansion in departures from the slow-roll approximation.
They can then be expressed in terms of observable quantities, such as the
square of the ratio of the gravitational wave amplitude to the density
perturbation amplitude, the deviation of the spectral index from the
Harrison--Zel'dovich value, etc. Here, we calculate complete expressions for
the second-order contributions to the coefficients of the expansion by
including for the first time corrections to the standard expressions for the
perturbation spectra. As well as offering an improved result, these corrections
indicate the expected accuracy of the reconstruction. Typically the corrections
are only a few percent.
