We show 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 μHz 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.