SUMMARY We shbw how to calculate exact complex eigenfrequencies and eigenfunctions and exact single mode synthetic seismograms for a spherical anelastic earth model. The real frequencies of oscillation of some spheroidal modes differ by 1 to 2 pHz from the corresponding spherical elastic earth eigenfrequencies. The decay rates are generally well approximated by conventional first-order perturbation theory. The complex radial eigenfunctions of the most strongly affected modes differ substantially from the corresponding real elastic eigenfunctions, and this can lead to significant perturbations in the initial phase and amplitude of the associated free oscillations following an earthquake. Most, but not all, of the strongly affected modes have large displacements in the inner core. Long-period synthetic seismograms on a spherical, non-rotating, elastic, isotropic (SNREI) earth are commonly calculated by summing the spheroidal and toroidal free oscillations. Anelasticity is generally accounted for by letting each mode decay at a rate determined by first-order perturbation theory; perturbations to the elastic radial eigenfunctions are ignored (Gilbert 1970). This intuitively appealing approach neglects the possibility of mode coupling due to anelasticity. In this paper we show how to calculate the exact complex eigenfrequencies and eigenfunctions of a spherical, non-rotating, anelastic, isotropic (SNRAI) earth model, and how to calculate exact synthetic seismograms on such an earth model. We adopt a coupled mode approach in which every SNRAI radial eigenfunction is expressed as a complex linear combination of the SNREI radial eigenfunctions; anelasticity and physical dispersion are fully accounted for. Unperturbed SNREI modes that are closely spaced in frequency and whose radial eigenfunctions are similar may be strongly coupled by spherical anelasticity; this coupling causes small shifts in the real parts of the complex SNRAI eigenfrequencies relative to the unperturbed SNREI eigenfrequencies. The complex SNRAI eigenfunctions of strongly coupled modes can differ substantially from the corresponding real SNREI eigenfunctions, and this can lead to significant perturbations in the initial phase and amplitude of the associated free oscillations following an earthquake. To gain insight into the phenomenon of anelastic mode coupling, we first consider the elementary classical mechanics problem shown in Fig. 1. The system consists of two linear springs each with an attached mass m; the first spring has stiffness k and the second spring has stiffness k + Sk, where Sk > 0. In the absence of friction the two masses oscillate independently with angular frequencies w = (k/m)' and w + 60 = [(k + Sk)/m]'n. Let ul(t) and u2(t) denote the displacements of the two masses and let a dot denote differentiation with respect to time. Suppose that the first oscillator is impulsively excited at time t = 0, so that the initial conditions are U,(O) ='O, U2(O) = 0, li'(0) = 1, liZ(0) = 0.