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I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular subject, the better.

Background: For my diploma thesis, I need in particular to understand asymptotics of the Fourier coefficients for certain entire functions, so I need to study it fast, that is, more straightforward, well-structured theory without much "bla-bla", and less exercises... Nevertheless, I would like to learn the more general theory of Fourier analysis for complex/holomorphic functions as it has a great deal of applications in Analytic Number Theory, which is one of the subjects of interest to me.

Thanks in advance!

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  • $\begingroup$ PS: How do I make this community wiki? $\endgroup$ – M.G. Mar 8 '11 at 17:36
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    $\begingroup$ you flag it for moderator attention. And then one of us will ride to your rescue :-) $\endgroup$ – Willie Wong Mar 8 '11 at 17:46
  • $\begingroup$ Related: math.stackexchange.com/questions/4422/… $\endgroup$ – Aryabhata Mar 8 '11 at 17:49
  • $\begingroup$ Just to clarify: I am NOT interested in texts on general/abstract/real Fourier analysis. What I am interested in, is Fourier analysis for complex-valued functions defined on domains in the complex plane, in particular holomorphic functions. @Willie Wong: Thanks for mentioning it! :-) From MO I was kind of used to make my threads CW by myself :-) $\endgroup$ – M.G. Mar 8 '11 at 18:23
  • $\begingroup$ @ex-falso: That is why this question did not get any close votes... I only added that comment, so that you get a convenient link to that question on the Linked section on the right side of this page (that question would get a link to this question too). $\endgroup$ – Aryabhata Mar 8 '11 at 19:04
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The following references cover some close links between harmonic and complex analysis that may be suitable for what you need (such as Paley-Wiener theorems, Corona Theorems, etc):

  • Geometric Function Theory: Explorations in Complex Analysis by Steven Krantz

  • Bounded Analytic Functions by John Garnett

  • A Guide to Distribution Theory and Fourier Transforms by Robert Strichartz

  • Real and Complex Analysis by Walter Rudin

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