Abstract
A problem in perturbation theory of differential equations is described which leads to some interesting technical difficulties in asymptotics. The leading behaviour of the imaginary part of the perturbed eigenvalues is obtained in the difficult case when this is of a much smaller order of magnitude than the real part. Although relatively small, the imaginary part is of importance in estimating radiation losses in bent optical fibre waveguides, where it represents the rate at which energy leaks out of the core
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