I have recently finished calculus by Michael Spivak, and wanted to learn more about things like complex analysis talked about towards the end of this book.I was also very interested in learning more about concepts like analytic continuation. Which books, in order, would you recommend me to continue my studies in analysis, so that I can finally understand these concepts? I have heard highly of the books by Stein and Shakarchi Should I start with their books on real and complex analysis? Another option is real analysis by folland. Any suggestions would be incredibly helpful.
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is it Calculus on Manifolds? – janmarqz Apr 24 '21 at 18:09
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Maybe try volume 2 of Courant and John? – Ben Apr 24 '21 at 21:58
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2Calculus is like extra diluted diet coke of analysis, so I recommend learning analysis now that you're comfortable with the language. Tao's Analysis 1, for instance. You can learn about Black Magic..I mean Complex Analysis.. from Rudin's, say. – AlvinL May 01 '21 at 15:06
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I have read half of rudin’s principles of analysis, though could not bring myself to complete it. I found it a bit dry at times, and heard that lebesgue measure is treated better in other books. Is rudin’s complex analysis any better? Or should I learn it from stein and shakarchi? – A.G May 01 '21 at 15:22
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I actually had heard quite a bit about folland on this site, and had that in mind. Should I read that, or should I read calculus on manifolds first? – A.G May 01 '21 at 15:25
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I think it's good to supplement books like Folland (which are very concise and don't provide much commentary about the big picture) with books like The Calculus Gallery and Journey Through Genius. – littleO May 02 '21 at 06:06
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Also, there are several books which attempt to be more readable than books like Folland and Rudin. For example, there is Axler's new book Measure, Integration, & Real Analysis and the new book Real Analysis: A Long Form Mathematics Textbook For Complex Analysis, you might check out Visual Complex Analysis by Needham. Basically you should look at a lot of different books and see what connects with you. – littleO May 02 '21 at 06:08
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You can also ask on academia stack exchange – Тyma Gaidash May 02 '21 at 15:02
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On second thought, I would not recommend Rudin for undergraduate level students. Rudin's is rigorous, yes, but it can be overwhelming at times. If you're willing to make a tradeoff for readability, then for Complex Analysis you can go with Stein & Shakarchi, Tristan Needham or Ablowitz & Fokas.
For analysis you could look into Tao's Analysis 1&2.
AlvinL
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I second the suggestion of Tristan Needham. It offers a (fairly) elementary and intuitive approach, while also bringing you towards more advanced result. It is visual, so it will give you plenty of intuition along the way, which will help you to move on towards more advanced textbooks afterwards. It is definitely a good book to start with. – Jonah May 02 '21 at 06:34