On elementary complex variable theory applied to generalised sinc(p) and related integrals

Abstract

The elementary application of complex variable theory in an earlier paper [Hey, J.D. 2020. On elementary complex variable theory applied to sinc and related integrals. Transactions of the Royal Society of South Africa 75(3): 295–306] is extended by use of the generalised sinc(p) function defined below, in order to provide some interesting, additional insight into the behaviour of the Borwein integrals, which arise as simple consequences of Jordan’s lemma applied to Cauchy’s theorem. The present treatment is of physical interest in relation to the analysis of spectral line broadening by the electric fields of ion perturbers in laboratory and astrophysical plasmas. Finally, a result, stated as a student problem in the well-known treatise [Whittaker, E.T. & Watson, G.N. 1927. A Course of Modern Analysis, Ch. VI (4th ed.). Cambridge University Press], is formulated in more general terms with the aid of the sinc(p) function. The simplicity of application of the present analytical results is extensively illustrated by numerical tables.

Keywords:

• generalised sinc(p) function
• complex path integration
• Jordan’s lemma
• Cauchy’s theorem
• Borwein integrals extended

ACKNOWLEDGEMENTS

Warm thanks are due to Miss Lynne K. Hey for drafting the figures.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).