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My university offers $3$ complex analysis courses:

  • an introduction computational course
  • Chapters $1$ - $5$ of Ahlfors
  • a course covering compactness and convergence in the space of analytic functions, Riemann mapping theorem, Weierstrass factorization theorem, Runge's theorem, Mittag-Leffler theorem, analytic continuation and Riemann surfaces, and Picard theorems

I plan to take these three in the next few years. However, I am curious about complex analysis and would like to continue my studies after the introductory level. Which area(s) of complex analysis are worth studying currently i.e. several complex variables, operator theory, or differential equations? Which texts should I pursue after Ahlfors? I am studying physics also, so a text with rigorous results in quantum mechanics or other related fields would be nice :)

Thanks in advance for advice!

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Reproducing Kernel Hilbert Spaces are studied at several levels, and they cover topics like machine learning and complex function theory. This falls under the banner of Function Theoretic Operator Theory. Duren’s book on the Hardy Space is a good place to start as is Hoffman’s text Banach Spaces of Analytic Functions. Alger and McCarthy’s text on the Pick Theorem uses RKHSs explicitly whereas the former texts don’t mention them (even though the Hardy space is a RKHS).

That’s where I got my start anyways. Nowadays I work on Approximation theory with RBF kernels. I feel that the analytic function theory helped get me where I am, and branching out to machine learning was straight forward.

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  • $\begingroup$ Thanks for the recommendations! I will take a look at Hoffman's text as it sounds interesting. What is required of this book? Would Ahlfors and Rudin suffice? @Joel $\endgroup$ – coreyman317 Jun 19 '18 at 6:14
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    $\begingroup$ Rudin’s Text would be more than enough. In fact he also touches on the Hardy spaces a bit in his book. $\endgroup$ – Joel Jun 19 '18 at 13:35

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