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Best books and papers for classical "hard" analysis tricks

  • #1
Sauvain

Main Question or Discussion Point

It seems that most recently published analysis textbooks are written from a more abstract perspective, e.g., dealing with functions on general metric spaces or with spaces of functions. However, often solving a problem requires some "get your hands dirty" classical or "hard" analysis -- for example involving clever manipulations of inequalities, algebraic identities, substitutions, tricks to rewrite integrals in other forms, and so on. I have noticed that a lot of older analysis textbooks contain a lot of this kind of analysis. Some books that I think fit with what I have in mind are:

* Whittaker and Watson
* "Baby" Rudin
* Zygmund's book on trigonometric series
* Stromberg's book on classical real analysis

My question is, what are some of the best books or papers for learning this kind of analysis? More generally, what is the best way to learn this kind of analysis? A process that has worked fairly well for me so far has involved attempting to solve problems of this type and studying "hard analysis" proofs. I just wonder if anyone has any suggestions or experiences to share of learning this kind of math.

Also, it seems that this kind of mathematics used to be more of a central focus, for example in the younger days of G.H. Hardy. Other mathematicians/scientists who seem to be / hav quite good at hard analysis include Freeman Dyson, Lars Onsager, George Polya. If anyone has any others to add to this list, I would be interested in hearing them also. I guess I'd be interested in seeing a "hard analysis" hall of fame. Of course it would have to include Euler, Gauss, and so on.

Finally, if anyone has any favorite results or papers in this vein, feel free to mention them as well.
 
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Answers and Replies

  • #2
Demystifier
Science Advisor
Insights Author
10,590
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The books
Boros and Moll, Irresistible Integrals
Nahin, Inside Interesting Integrals

are full of tricks for dealing with integrals.
 

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