An analytic Green's function is derived for a lined circular duct, both hollow and annular, containing uniform mean flow, from first principles by Fourier transformation. The derived result takes the form of a common mode series. We show that the analytic Green's function for a lined hollow circular duct, containing uniform mean flow, is essentially identical to that used by Tester et al. in the Cargill splice scattering model. The explicit form of the Green's function for the annular duct is new. A more comprehensive causality analysis suggests the possibility of certain upstream modes being really downstream instabilities. As their growth rates are usually exceptionally large, including these modes as instabilities is both not practical and in disagreement with most (not all) experiments. Therefore, we outline the possibility but do not include them in the presented examples. We follow the "modelling assumption" that all modes decay in their respective direction of propagation. To illustrate the advantages of our analytic result compared to the matrix inversion technique of Alonso et al., we compute the mode amplitudes from both methods for a typical aircraft engine intake condition. The comparisons show good agreement without flow, irrespective of how many modes are included in the matrix inversion for the numerical mode amplitudes. With flow, the mode amplitudes do not agree but as the number of modes included in the matrix inversion is increased, enough to include any important surface waves, the numerically obtained modal amplitudes of Alonso et al. appear to be converging to the present analytical result. In practical applications our closed form analytic Green's function will be computationally more efficient, especially at high frequencies of practical interest to aero-engine applications, and the analytic form for the mode amplitudes could permit future modelling advances not possible from the numerical equivalent. It also may have application to post-processing of phased array measurements inside lined ducts.