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I have earlier self studied Tom M. Apostol's Introduction to Analytic Number Theory after doing a course in complex analysis, but my instructor at university didn't even mention analytic continuation. Although I self studied from Complex Variables with Applications from Ponnusamy and Silvermann and then studied Chapter 12 and 13 of Apostol's Introduction to Analytic Number Theory but I don't feel sometimes comfortable in analytic continuation.

Can you please suggest some good reference book for Analytic Continuation which has explained analytic continuation in detail and also contains exercises based on analytic continuation which I can try?

  • Your question is weird because the main theorems about $\zeta(s)$ (and $\Gamma(s)$) are a very good motivation and application of the main theorems in complex analysis. – reuns Dec 09 '19 at 15:34
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    @reuns if you can tell a book which covers analytic continuation in detail and has examples, it would be of great help. –  Dec 15 '19 at 05:57
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    I would like to mention that as this question was not answered back then so I read the chapter on analytic continuation again from Ponnusamy and Silvermann Complex Variables with applications and was able to follow it and it improved my understanding, also I read chapter on Analytic continuation from Serge Lang's Graduate Text in Complex analysis. For the questions I had in Apostol's Introduction to analytic Number Theory in analytic continuation of functions I asked here on MSE. I am glad that user: upanddownintegrate put an bounty on it because ... –  Sep 16 '21 at 08:17
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    ...i still think if people knew about a book which has really good explanation of concepts of analytic continuation along with good number of quality exercises it will really help people who are new to this concept or are struck on it as it is very important for analytic number theory. –  Sep 16 '21 at 08:18

1 Answers1

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The most complete monograph on analytic continuation is perhaps still the more-than-70-years old monograph of Bieberbach [1], which is entirely (and exclusively) devoted to the subject. It deals extensively with various known continuation theorems, which give (usually sufficient) conditions for a power series to be continuable across the boundary of its convergence disk (including a necessary and sufficient condition for a point to be non-regular, i.e. to be a point across which it cannot be analytically continued, due to Fabry), and with almost all the known methods for analytically continuing power series (including but not limiting to Borel-Laplace and Laplace transforms). Also it gives very detailed historical context for the development of the subject. However, it has a few evident defects namely

  1. It is written in German (and this perhaps is not really a serious flaw) and
  2. It is outdated in some of his sections (time has passed even for this great classic), and finally (perhaps the worst one)
  3. It does assume the basic concepts are known, thus it does not develope the topic from scratch.

Said that, a treatment that is not as comprehensive but builds up from the basics up to intermediate levels of difficulty and offers also several exaples and exercises, can be found in the books of Markushevich, particularly [3], chapter 9 ("Analytic continuation"), but also [2], chapters 16 and 17 ("Power series: rudiments" and "Power series: ramifications") of volume I and chapter 8 ("Analytic continuation") of volume III. The exposition starts from basic conceps i.e. singularities of analytic functions, the convergence disk of a power series (including the same necessary and sufficient condition for a point to be non-regular given by in [1]), the meaning of an analytic element, etc. and goes up to giving an elementary introduction to the Borel-Laplace transform.

My advice

My advice is to use the textbooks of Markushevich to get aquainted to the topic at a deeper level than other textbooks allow, and use the book of Bieberbach as a sourcebook when you need to go to further deeper. Indeed, even if you don't understand German very well, you can use [1] as a guide to the many English or French language original papers: I follow this route almost always.

References

[1] Ludwig Bieberbach, Analytische Fortsetzung (German) Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 3, Berlin-Göttingen-Heidelberg: Springer-Verlag, pp. IV+168 (1955), MR0068621, Zbl 0064.06902.

[2] Alekseĭ Ivanovich Markushevich, Theory of functions of a complex variable. Vol. I. (English), Selected Russian Publications in the Mathematical Sciences, Englewood Cliffs, N.J.: Prentice-Hall, Inc., pp. XIV+459 (1965), MR0171899, Zbl 0135.12002, and
Theory of functions of a complex variable, Vol. III. (English), Selected Russian Publications in the Mathematical Sciences, Englewood Cliffs, N.J.: Prentice-Hall, Inc., pp. xi+360 (1967), MR0215964, Zbl 0148.05201.

[3] Alekseĭ Ivanovich Markushevich, The theory of analytic functions: a brief course, translated from the Russian by Eugene Yankovsky (revised from the 1978 Russian edition) (English), Moscow: Mir Publishers. pp. 423 (1983), MR0708893, Zbl 0499.30002.

Daniele Tampieri
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