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
where the Laurent coefficients an
have the integral representation
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
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.