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I'm looking to brush up my analysis after several years out of mathematics (graduated in 1996 and worked in industry since then).

I want a hard core treatment of analysis in order to brush up and progress in analytic number theory.

Bromwich's infinite series looks really good to me, though I'm concerned about his treatment of divergent series and possibly other topics being somewhat dated or might take me on the wrong track as a result.

Other than that the book looks excellent, but would you the community recommend using this ? or possibly you can recommend a modern counterpart ? or perhaps some combination ?

I work alone so helps not to have to change a book half way through finding that it's misleading / dated, hence I ask for your advice.

PS The level of the treatment would correspond roughly to graduate level these days to give you a better idea.

PPS Ideally you've read or know this book already, if not then what I'm looking for is this : A 'brisque' refresher of convergence / uniform convergence topic plenty of example problems moving rapidly onto such topics as e.g. Borel's integral, Pringsheim's theorems, Tauberian sums, good treatment of theta functions etc

As I say I want later to go on with analytic number but want I nice broad treatment as a thorough refresher with plenty of problems in it. I'm concerned since the book is dated that there may be subtleties in the books that may be misleading these days for later more serious work.

Thanks,

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  • $\begingroup$ What level are you looking for? Graduate school level or what? $\endgroup$
    – David Roberts ♦
    Commented Jun 9, 2016 at 14:41
  • $\begingroup$ "Hard core treament of analysis" seems very broad to me, especially since you then say "graduate level" and are apparently seeking to "progress in analytic number theory". What topics do you want covered, and in what depth? $\endgroup$
    – Yemon Choi
    Commented Jun 9, 2016 at 15:18
  • $\begingroup$ If you're looking for something from the same era, Whittaker/Watson's A Course of Modern Analysis comes to mind. I wasn't sure how relevant it would be to analytic number theory, but when I did a google search for {Whittaker Watson "number theory"}, these notes by Noam D. Elkies showed up on the first page of hits. $\endgroup$
    – Dave L Renfro
    Commented Jun 9, 2016 at 19:43
  • $\begingroup$ I'm a little late, but I'd highly recommend Bromwich, perhaps with a supplemental text if you are paranoid about it being out of date. What I found reading Bromwich is that all of the material is the same but the explanations are much more "old school", which I actually found helped me. $\endgroup$
    – JMJ
    Commented Dec 13, 2018 at 13:05

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Concerning the depth of treatment of convergence of series, Bromwich's book looks quite good and thorough despite being old. Plenty of example problems can be found in http://www.maa.org/press/books/real-infinite-series (Real Infinite Series, by D.D. Bonar and M.J. Khoury) and http://www.maa.org/press/ebooks/excursions-in-classical-analysis (Excursions in Classical Analysis, by H. Chen). On divergent series there is G.H. Hardy's classic "Divergent series", however it is quite tough.

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