  A straight cylindrical duct is considered containing an axial mean flow that
is uniform everywhere except within a boundary layer near the wall, which need
not be thin. Within this boundary layer the mean flow varies parabolically. The
linearized Euler equations are Fourier transformed to give the Pridmore-Brown
equation, for which the Greens function is constructed using Frobenius series.
Inverting the spatial Fourier transform, the critical layer contribution is
given as the non-modal contribution from integrating around the continuous
spectrum branch cut. This contribution is found to be the dominant downstream
contribution to the pressure perturbation in certain cases, particularly for
thicker boundary layers. The continuous spectrum branch cut is also found to
stabilize what are otherwise convectively unstable modes by hiding them behind
the branch cut. Overall, the contribution from the critical layer is found to
give a neutrally stable non-modal wave with a phase velocity equal to the mean
flow velocity at the source when the source is located within the sheared-flow
region, and to decay algebraically along the duct as $O(x^{-5/2})$ for a source
located with the uniform flow region. The Frobenius expansion, in addition to
being numerically accurate close to the critical layer where other numerical
methods loose accuracy, is also able to locate modal poles hidden behind the
branch cut, which other methods are unable to find; this includes the
stabilized hydrodynamic instability. Matlab code is provided to compute the
Greens function.
