{"title":"On elementary complex variable theory applied to generalised sinc(p) and related integrals","authors":"J. Hey","doi":"10.1080/0035919X.2021.2005716","DOIUrl":null,"url":null,"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.","PeriodicalId":23255,"journal":{"name":"Transactions of The Royal Society of South Africa","volume":"77 1","pages":"11 - 25"},"PeriodicalIF":0.0000,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of The Royal Society of South Africa","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/0035919X.2021.2005716","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Agricultural and Biological Sciences","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍：
Transactions of the Royal Society of South Africa , published on behalf of the Royal Society of South Africa since 1908, comprises a rich archive of original scientific research in and beyond South Africa. Since 1878, when it was founded as Transactions of the South African Philosophical Society, the Journal’s strength has lain in its multi- and inter-disciplinary orientation, which is aimed at ‘promoting the improvement and diffusion of science in all its branches’ (original Charter). Today this includes natural, physical, medical, environmental and earth sciences as well as any other topic that may be of interest or importance to the people of Africa. Transactions publishes original research papers, review articles, special issues, feature articles, festschriften and book reviews. While coverage emphasizes southern Africa, submissions concerning the rest of the continent are encouraged.
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