Numbers and Functions: From a Classical-Experimental Mathematician’s Point of View (Student Mathematical Library) by Victor H. Moll (PDF)

Description

New mathematics often comes about by probing what is already known. Mathematicians will change the parameters in a familiar calculation or explore the essential ingredients of a classic proof. Almost magically, new ideas emerge from this process. This book examines elementary functions, such as those encountered in calculus courses, from this point of view of experimental mathematics. The focus is on exploring the connections between these functions and topics in number theory and combinatorics. There is also an emphasis throughout the book on how current mathematical software can be used to discover and prove interesting properties of these functions. The book provides a transition between elementary mathematics and more advanced topics, trying to make this transition as smooth as possible. Many topics occur in the book, but they are all part of a bigger picture of mathematics. By delving into a variety of them, the reader will develop this broad view. The large collection of problems is an essential part of the book. The problems vary from routine verifications of facts used in the text to the exploration of open questions.

User’s Reviews

Editorial Reviews: Review On the whole, this is an extraordinarily interesting book overflowing with (mostly) elementary non-routine mathematics. It’s well-written and a pleasure to read. I’ve been keeping it on my desk for the ease of access; it’s going to stay there for some while. I recommend it wholeheartedly to math instructors, teachers, and students, especially those who have only a slight interest in the subject. The book is bound to expand their horizons. –MAA Reviews

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

⭐Victor Moll, one of the greatest experimental mathematics researchers of our time, is also a very gifted teacher.Luckily, he wrote up the lecture notes for his innovative classes in experimental mathematics, so thatundergraduate math (and science!) majors can get a glimpse of mathematics that is so much more funthan the same old lemma/theorem/proof/corollary drivel that turned off so many talented people away frommathematics, not because they were not capable of mastering it, but because it was no fun.Exploring mathematics the way Moll does, by experiment, has the potential to attract to math all thesevery talented young minds. For the math prof, it should be a case study, and paradigm, of howto teach math the fun way!

⭐Mathematical texts serve a variety of aims. A text like Rudin’s Analysis is held together by a narrative that is essentially singular: you learn the theory of Analysis from the author’s vantage, starting at the beginning. You may do some applications here and there, but it is a long work dedicated to justifying the standard tools. So you get the rudiments of a theory that requires laying a lot of brick. Numbers and Functions is not a text like this. Instead, each chapter breaks into a topic, highlights results and problems, and lays everything out very clearly, before moving to another topic. These topics are all related, but in a manner that is non-trivial. I find the style succinct and exciting. Anyway, to start from first principles and then arrive at all these topics would require thousands of pages, so the reader should come with some background. I love this book. It is not for everyone–nothing is universal. But if you are interested in classical analysis, special functions, integration, number theory, symbolic computing, and some other deep topics like the AGM, this will grab you by the collar and put you to work. Very well done.

⭐There is no coherence to this book. The helter skelter style veers from onesubject to the next with no central goal or underlying plan. No sooner is atopic picked up than it is discarded and it’s on to something completelydifferent. There are no solutions to the exercises and no reasons toown a copy of this book.

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