I am currently doing a deep research in the theory and applications of infinite series, particularly power series, trigonometric series and numerical series. The following books are my collections. They reflect my own personal opinion and thus are influenced by my own inept mathematical ability. Most of these books can be downloaded and read here: http://gen.lib.rus.ec/
1) The rise and development of the theory of series up to early 1820-Giovanni Ferraro
This is perhaps the only monograph on the history of infinite series (numerical, power and trigonometric series) in English so far. The author, judging by the name, must be an Italian historian. The work does offer a compendium of facts on the origins of the development of certain types of series found in the work of XVII to early XIX century. The book, however, suffers from typos, disorganization, and fragmented presentation. For example, it offers to introduce the concept of generating function (les fonctions généatrices) found in Laplace's "Théorie analytique de probabilités", a pioneering work in probability. But what the author does is just citing verbatim the original text without giving much explanations on the meaning, structure and importance of this development. It is quite a shame. But what can on expect from a book of 408 pages. The history of infinite series, if carefully studied as in the case of trigonometric series (see Paplauskas book below), will take up much more spaces than just this volume, albeit rather thick already for a publication of this sort.
2) James Stirling's Methodus Differentialis-Ian Tweedle
Sách này là bản dịch lần thứ hai từ tiếng Latin sang tiếng Anh. Sách là kho của nhiều phương pháp và cách thức biến đổi chuỗi. Trong sách có cả công thức tính xấp xỉ Stirling, chuỗi asymptotic của logarit hàm Gamma. Phương pháp trong sách hơi khó đọc nhưng vẫn ứng dụng được vào Toán học ngày nay. Nhìn chung nội dung sách có nhiều chỗ liên quan đến giải tích số (numerical analysis), Mình đã tìm được dạng đóng cho hai chuỗi lũy thừa trong sách này. Đây là thành quả mà mình xem là tuyệt nhất kể từ khi mình bắt đầu học Toán.
3) Introduction to Analysis of the Infinite-Leonard Euler, dịch bởi John D.Blanton hay Ian Bruce
Needless to introduce, this book is definitely a classic that people who love history of mathematics like me cannot forget to read. Euler, along with Ramanujan, still remained the ultimate master of infinite processes that are unsurpassed by anyone. Despite being one of the greatest mathematicians of his age, Euler wrote in a very careful, simple, yet insightful pedagogic style. For example, he was able to present the Maclaurin series of $\sin(x)$ and $\cos(x)$ without using the definition of differentials and differentiation. He also included his remarkable results of summing infinite series of reciprocal integers up to the 26th power. The famous French mathematician, who lived in the 20th century, Andre Weil, said like this, according to Blanton's quote:
""our students of mathematics would profit much more from a study of Euler's Introductio in analysin infinitorum, rather than of the available modern textbooks."
Another acclaim for the book is here:
The eminent historian of mathematics, Carl Boyer, in his address to the International Congress of Mathematicians in 1950, called it the greatest modern textbook in mathematics. Boyer cited Euclid’s Geometry as the greatest mathematical textbook of the classical period, perhaps of all time, appearing in over one thousand editions. For the medieval period, he chose the less well-known Al-Khowarizmi, largely devoted to algebra. But for “modern” times, Boyer made the case for Euler’s Introductio as the greatest modern textbook — and, appropriately, this time a text in analysis.
One of the remarkable difference between this book and modern math books is the layout. Euler didn't present theorems, corollary, lemma, proofs in the style of modern textbooks, nor theorem and demonstrations, a style mimicking "the Elements" of Euclid, which is common in his time. He just showed you all kinds of algebraic manipulations, change of variables (called transformation), and a long list of calculations he made to derive the final result. He was indeed a human calculator with unrivaled memory. I find his book highly interesting and insightful.
4)Infinite series-Earl D.Rainville
I like this book very much. It is a gentle introduction to the theory of infinite series, written by an American mathematician who wrote many other books on differential equations. The style of presentation and notation is fully modern. It contains a lot more than what I expect from an elementary treatise. It contains normal topics found in Calculus 2's curriculum in the USA, including sequences, numerical series, power series, conditional and absolute convergence, many kinds of tests for convergence. Uniform convergence receives a chapter, emphasizing its importance. The author provides both formal definitions and graphical interpretation, which is very helpful. The shining chapters of this books are special functions defined by series such as Gamma, Beta, Hypergeometric functions. Then it has an elementary chapter on Fourier series, asymptotic series, infinite products. The topic of accelerating convergence, which is very rarely mentioned in a Calculus course, is given an elementary treatment here.
I wasn't able to appreciate this book because when I bought it, I was still very clumsy at manipulating sigma notations. With practice and constant exposure, I was able to get much better, which means I could read the book with more ease and understand various derivations and manipulations that the author made. I highly recommend this book for those who share a similar interest in infinite series like me.
5)Traité élémentaire des series-Eugene Catalan
I used to think highly of this book. I owned 2 copies, one I already gave it to a friend of mine, Trong Dat Do. I recently bought a new one with a reasonable price. The author was a prominent mathematician who worked mainly in combinatorics and number theory. The Catalan numbers bear his name.
The reason I change my initial positive opinion of this book is because, as my mathematical knowledge enlarges, I find this book quite hard to read. Its layout is not very logical, and the scope is rather limited. Its strong point is that it contains many (sometimes artificial) examples of series. It does contain a lot more series than many other books listed here, and some are cleverly deployed to show the ability of its author. At the end of the book, there are many trigonometric series that are rarely found in other books, so it is not at all a bad book. It is just that I don't highly rank it as before.
6) Introduction to the theory of infinite series-Bromwich
This book is theory heavy. I used to own a newer copy printed in 1926, but I sold it when I moved from Philadelphia to Ohio. Then I rebought an older copy which was printed in 1908. I liked this version more as I felt the author had written more clearly before he decided to add new stuffs, which make the text look cluttered and hard to follow. The book has ample challenging exercises (many extracted from the old Tipos exams) that really intimidate me. It shows how demanding the curriculum was in England at the beginning of the 20th century. This reminds me of the book "A Course in Pure Mathematics" by Hardy. But it covers lots of topics that are hard to find in other treatises.
There are some problems such as: in the table of contents, the chapter on power series, it says that there is a section on differentiating and integrating power series. I thought that I could find some theorems on the unchanging of radii of convergence of power series when differentiating and integrating terms by terms, and the worse of all can only be the loss of the two terminal ends of the radius of convergence. But when I flipped to that section, there is nothing. The section entitled differentiating and integrating power series is left to the exercises, which I find rather irritating.
The book has a chapter on non-converging (which is divergent in other works) and asymptotic series. I am quite interested to know more about asymptotic series because divergent series are said to be banished from mathematics since Abel and Cauchy imposed rigors on this field. It was Henry Poincare who reintroduced these series into his work entitled "New Methods of Celestial Mechanics: Periodic and Asymptotic Solutions".
It is a solid book to own, and after letting your fear subside, find and try a few simple exercises to improve your skills.
7)Theory and application of infinite series-Knopp
I no longer own this book. It is quite similar to Bromwich's book, as shown above. It doesn't have lots of exercises like Bromwich, but it contains a very long, detailed, almost philosophical approach to construct the system of real numbers. Judging from the length of this section, the author must place a significant emphasis on its importance. It is indeed a rather dense and difficult book. On the section of "multiplying series", the author offers many theorems and include proofs
Quyển này cũng là một quyển khô khan chứa đựng nhiều lý thuyết hơn thực hành. Mình cũng ít đọc nhưng đôi lúc gặp phải chuỗi logarit thì mình đọc phần có liên quan. Sách cũng đáng đọc nếu bạn có trình độ thâm sâu hơn mình. Sách viết khá đầy đủ và có nhiều phần khó tìm trong những sách khác.
8) Methods of solving sequence and series problems-Ellina Grigorieva
I no longer own this book. It is a book specifically designed to train and inspire those who take the Olympiad Competitions. It contains some nice chapters on the application of calculus to summing series. But this book is not comprehensive and I sold them online.
9) Паплаускас А.Б. Тригонометрические ряды от Эйлера до Лебега (Paplauskas- Chuỗi Lược Giác Từ Euler đến Lebesgue) (Paplaukas Trigonometric series from Euler to Lebesgue)
My Russian is fairly limited. I am struggling with cases and conjugating verbs, but I love this language very much. I have learnt quite a lot of mathematical terminologies by reading the 3-volumes work of Grigori Fichtengolz on analysis, so I can understand some bits of this book. It is a very rare and very important book because it recounts the birth and growth of the theory of trigonometric series and Fourier series. It is sad that there is no translation in English. It should suit those who want to understand more about the profound changes mathematics have gone and still go through when trigonometric series were studied by 19th century mathematicians. I don't need to go into further details by reminding readers that long generations of mathematicians devote all their brain power on figuring out various aspects of trigonometric series. We have Fourier, who excellently derived the coefficients of a Fourier cosine and sine series. We then have Dirichlet, Riemann, Georg Cantor, Thomae, just to name a few prominent mathematicians in the 19th century. In the 20th century, we continue witnessing the growth of this field, with Kolmogorov becoming famous for his construction of a Fourier series that almost diverges everywhere, then diverge everywhere. Then, we have Arnaud Denjoy with his full works entitled "Leçons sur le calcul des coefficients d'une série trigonométrique", Antoni Zygmund seminal work "Trigonometric Series", Nina Bary "A Treatise of trigonometric series". Harmonic analysis now become a separate branch of mathematics and continue to receive steady attentions from modern scholars. All these developments are traced in this book by Paplaukas. It is a very important work in my opinion.
I have tried to look up the name Paplauskasa as I was curious about the author, both in English and Russian, but I could find very little details about him. He seemed to be a professional historian of mathematics who worked in the Institute of History of Natural Science and Technology (институт истории естествознания и техники). I have strong admiration for various Russian school of mathematics. They have a very original tradition of doing and researching math, and in this case, even history of mathematics.
10) Воробьев Н.Н. Теория рядов (Vorobev N.N Lý thuyết chuỗi)
This book is written with a very careful style. I have only been able to progress through the first two chapters. It makes a clear distinction between sequence of numbers and sequence of functions. It defines the radius of convergence for series of functions. The book area includes:
1) Sequence
2) Numerical series
3) Functional series (power series and Taylor series)
4) Functional series
5) Fourier series
6) Double series
7) Double series
8) Summing convergent series
9) Summing divergent series (including Poisson-Abel summing method, Tauber theorem, Euler method)
10) Convergence of Fourier series (a difficult topic)
As the author wrote in his preface, this book is a textbook designed for specialized course in engineering-technical school in the now defunct USSR. It is typical of Russian pedagogical style, stay away from heavy formalism but maintain rigor. It is indeed a wonderful book to have.
11) Georgi P.Tolstov Fourier Series
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