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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!

  • 0
    For clarification: what exactly do you mean by Fourier analysis for holomorphic functions? Are you talking about taking Fourier transforms over particular slices?2011-03-09

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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