Nine Introductions in Complex Analysis by Sanford L. Segal (PDF)

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Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The only two English language books I’ve found that present the Fabry gap theorem are this one and Montgomery’s “Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis”. Suppose that a power series converges has radius of convergence 1. This power series defines an analytic function on the open unit disc, and we want a criterion based on the coefficients of the power series whether the function can be analytically continued outside of the open unit disc. Fabry’s gap theorem shows that if the nonzero terms in the power series are sparse enough, then the unit circle is a natural boundary of the analytic function. Although the material on gap theorems is in the sixth chapter, it does not depend on the previous chapters and so can be read by itself.This is not a book to first learn complex analysis from. Depending on how sophisticated a presentation one wants, I would use Stein and Shakarchi’s “Complex Analysis”, Henri Cartan’s “Elementary Theory of Analytic Functions of One or Several Complex Variables”, or Walter Rudin’s “Real and Complex Analysis”.

⭐I know for a fact that Mary Anne Maher PhD contributed a great deal of very hard work to this text but is not mentioned as she was in the first edition.

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