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The problem of determining the eigenvalues of an inhomogeneous waveguide, subject to boundary conditions at an interface with an exterior homogeneous medium (cladding) is considered. A recently developed asymptotic method is summarised. Previous work dealt with planar waveguides; here we incorporate the extensions necessary to accommodate cylindrical waveguides, as encountered in optical fibres. The method uses a nonlinear transformation between variables to reduce a given differential equation to canonical form; this transformation has more tractable asymptotic properties than the original differential equation.