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Comment by graycat | Hacker News Reader

Suggestion: Set aside complex variables and do measure theory instead. For the Fourier and Laplace transforms, cover those via measure theory. For measure theory, H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis.

Then take a course in graduate probability which is based on measure theory. Good authors are Loeve, Neveu, Breiman, Chung, among others. So, learn about the cases of convergence including almost sure convergence, a good version of the central limit theorem, proved carefully, the weak and strong laws of large numbers, ergodic theory, and martingale theory.

Learn stochastic differential equations.

Learn some potential theory.

Then do some work in optimization and optimization under uncertainty.

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Personally, I think complex analysis is important. I really like Serge Lang's book "Complex Analysis." It is supposedly a graduate level text, but the first half of it covers the basics at a level suitable for upper division undergrad class. Ahlfor's book is also really really good, but a little harder to read in my opinion.

I didn't like either Royden or Rudin. I really like Folland's book "Real analysis: modern techniques and their applications." His Fourier analysis book is also pretty good IMO.

Yes, I've got a copy of Alfors. And at one time I started a course from a student of Alfors. And I have a book by Hille.

I just never could see much utility in complex variables: Complex valued functions of real variables. Sure. Complex valued measures? Sure. Fourier theory making a lot of use of complex numbers? Of course. A vector space where the field is the complex numbers? Certainly. Hermitian and unitary matrices with complex numbers? About have to like those due to the fundamental theorem of algebra, that is, roots of polynomials and, thus, complex eigenvalues. Functions of a complex variable? Never could see the utility.

Incredible, thank you for the guidance! I'll look into these topics more carefully

Complex valued functions of a real variable are common and important but well covered in real analysis. Functions of a complex variable I've never seen in practice.

A good course in statistics, e.g., with sufficient statistics, needs graduate probability, and with a grad probability course can do elementary statistics easily in the footnotes. E.g., the proof I worked out for the Neyman-Pearson lemma is quite general but based on the Hahn decomposition based on the Radon-Nikodym theorem (Rudin gives von Neumann's cute proof) in measure theory. The grown up approach to conditional probability is based just on the Radon-Nikodym theorem of measure theory.

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