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The Asymptotic Behaviour of the Laurent Coefficients

Published online by Cambridge University Press:  20 November 2018

Max Wyman
Max Wyman*
Affiliation:
University of Alberta
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 534 - 555
Copyright
Copyright © Canadian Mathematical Society 1959

References

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