The Asymptotic Behaviour of the Laurent Coefficients

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Abstract
Let G(z) be a function of a complex variable, regular in the annulus 0 ≤ a ≤ |z| < b ≤ ∞. We shall assume there exists a curve within the annulus for which provided z is restricted to be a point of this curve. Under these restrictions G (z) has a Laurent expansion of the form 1.1 where the Laurent coefficients an have the integral representation 1.2 and C can be any contour, within the domain of regularity, that encloses z = 0. We shall also assume that the an are all real numbers. Using the usual complex conjugate notation, we can, therefore, write 1.3 The problem of determining the asymptotic behaviour of an as n —> co is very old in mathematical literature and appears in many forms and disguises.

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