Cauchy and the Creation of Complex Function Theory free ebook download.

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Cauchy and the Creation of Complex Function Theory free ebook download

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Mathematics

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Author(s):	Frank Smithies

Date:	2008	Format:	djvu	Language:	English	ISBN/ASIN:	0521068878
Pages:	224	OCR:		Quality:		ISBN13:
Uploader:	eternal	Upload Date:	3/20/2024 5:43:40 AM

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0521068878 djvu
In chapter 1 we look at what was done before Cauchy. Euler and Laplace and others saw that complex changes of variables was a useful technique for evaluating real integrals. This is rather mysterious and it was treated with some suspicion. Cauchy set out to justify and systematise such techniques in his 1814 memoir (chapter 2), and then he kept polishing his results over the next ten years (chapter 3). Then comes his watershed 1825 memoir (chapter 4). Cauchy has now realised that all of the above should be understood in the context of path integration in the complex plane. Here integration is largely determined by the poles, prompting a calculus of residues, which he develops over the next couple of years (chapter 5). Another area of classical analysis where the complex viewpoint proved essential was the convergence of series (chapter 6). Cauchy's starting point here was the Lagrange series, first employed fifty years earlier by Lagrange, e.g. in celestial mechanics, without regard for its dubious convergence properties.

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