We give a systematic formulation and a rigorous justification of a perturbation technique for the computation of the eigenvalues and eigenfunctions of Love waves (and toroidal oscillations by an appropriate change for variables) in an anelastic medium with a constitutive law modelling geophysical media of current interest such as the Kelvin-Voigt Solid, the Maxwell Solid, the Standard Linear Solid, and the Standard Linear Solid with a continuous spectrum of relaxation times. We develop expressions relating the eigenvalues of eigenfunctions for Love waves in a continuously varying vertically stratified anelastic half-space to the corresponding elastic eigenvalues and eigenfunctions. Analytically, our correspondence principle has the form of a regular perturbation expansion in terms of a parameter E for both the eigenvalues and eigenfunctions. The identification of E is motivated by the dissipativity principle of viscoelasticity theory. Moreover, we show that our correspondence principle applies respectively only in the high and low frequency range for the Maxwell and Kelvin-Voigt Solids. Outside of the applicable range of frequencies, our correspondence principle yields no useful information. For the family of Standard Linear Solids it is uniformly applicable for all non-zero frequencies.We also derive an explicit formula to estimate the radius of convergence of our perturbation expansions. This estimate of the radius of convergence for each eigenvalue and eigenfunction is functionally defined by the constitutive model for the anelastic medium. The estimate is frequency dependent and depends on the separation distance between the eigenvalue and the remainder of the spectrum of the corresponding elastic problem.The stress-strain constitutive relationship for an anelastic medium is derived in Christensen (1971), Leitman & Fisher (1973), or Pipkin (1972. It is given by is the classical infinitesimal strain tensor. Gijkl(x, t ) has the representationWe will restrict our discussion to isotropic media. In this case (Christensen 1971), Gjjkz(x, t ) = 7 3 { G z (~, t ) -G I (X, t ) ) 6ij6kl' % G I ( x , t)(&ikajZ + 6iZbjk)( 2.4) where G l ( x , t ) is called the shear relaxation function, G z ( x , t ) is called the longitudinal relaxation function, andh j j is the Kronecker symbol. The relaxation functions Gi, i = 1, 2, are represented by Gi(t) = Gi(0) + IOt Gl(7)dT i = 1 , 2 , (2-5) where the real constants G i ( 0 ) are called the instantaneous moduli. They are related to the elastic response of the medium. If G,(O) = 0, i = 1 , 2 , then the medium has no instantaneous elastic response. If Gi(-) = lim G,(t) exist, then Gi(m) are called the equilibrium moduli. The momentum equation (2.1) and the constitutive relation (2.2) form a complete system of equations when supplemented with appropriate initial and boundary conditions (Aki & Richards 1980). Next, we derive the equation for Love waves in a continuously varying vertically stratified anelastic half-space. Let i = 1 , 2 , t-+ -5 2 = { ( x , z ) : -m < x < m , z 2 O } ...