[UPDATED] lovely book for the beginning student to understand analysis
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Hi [updated]
Whilst trying to get around my problem in understanding 'Residues' whist studying 'Complex Analysis' by Stewart, i returned to this book for help. Its true the type-face is small and there are very few graphs in-between the reams of information.
The book is written in a semi-formal manner, but the level it's pitched at is straight-forward if you have read these topics from studies elsewhere. The nice thing about this it helps to compile these topics in my mind from a newer, more formal perspective.
The way he writes you can tell he's an enthusiast for the topic. And its still great value for money!
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Getting started in math analysis
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This book by Shilov covers the fundamentals in beginning analysis(both real and complex). It has in common with Walter Rudin's book (entitled 'Real and Complex Analysis') that it covers both real functions (integration theory and more), as well as Cauchy's theorems for analytic functions. Shilov's book is at an undergraduate level, and it can easily be used for self-study. The Dover edition is affordable. Rudin's book is for the beginning graduate level, and it is widely used in math departments around the world. Both books have stood the test of time.
Comparison of Shilov with Rudin: Rudin's 'Real and Complex' has become an institution, and I have to admit I have loved it since I was a student myself, but conventional wisdom will have it that Shilov is a lot gentler on students, and much easier to get started with: It stresses motivation a bit more, the exercises are easier (some of Rudin's exercises are notorious, but I find the challenge charming--not all of my students do though!), and finally Shilov gets to touch upon a few applications; fashionable these days. But that part easily gets dated. I will expect that beginning students will enjoy Shilov's book.
Personally, I find that with perseverance, students who keep at it with Rudin's book, will end up with a lot stronger foundation. They are more likely to have proofs in their blood. I guess Shilov can always serve as a leisurely supplementary reading to Rudin.
There will never be another book like Rudin's 'Real and Complex', just like there will never be another van Gogh. But the fact that we love van Gogh doesn't prevent us from enjoying other paintings.
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Dr. Shilov could teach complex analysis to a chimpanzee.
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The book is one of the finest mathematics texts that I have ever read. That does'nt mean much because I'm 16 years old, and have only been studying advanced math for a little over a year. The treatment was basic enough for someone like me to understand. I strongly recommend it to college students and young people who are interested in mathematics. After introducing concepts from set theory, the foundations of analysis, and the notion of a "mathematical structure," he gives a detailed presentation of limits and series. He also introduces elementary functions in terms of their functional equations. Then he covers differentiation and integration, first of real, and then of complex functions. He uses Taylor series to introduce ideas about complex functions. In short, it is a good book for those who hope to become mathematicians, physicists, or engineers, and have had a few college math courses already.
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