# The Real and the Complex: A History of Analysis in the 19th Century

This is a history book on the development of mathematics in the 19th century. Each chapter is built up around one or a few mathematicians. First a short bio is sketched, often embedded in the political context of their time, but the more important part is where it is shown what theory the person has developed, in what context it was done, hence why he (there are unfortunately no she's) did it. It shows that even the big shots of mathematics that contributed to the tremendous expansion of mathematical knowledge in the 1800s were searching, sometimes uncertain or even mistaken. With several of the emerging concepts that we are now familiar with today still being shaped and reshaped, these were first formulated, not taking care of the tiny details, which needed a revision or even rethinking the concept later on. It was also the century in which analysis came to the foreground as one of the main mathematical topics next to geometry and algebra that had dominated before. Analysis became a working tool that was used in other domains of mathematics, it became essential in modeling physical phenomena and it was intensively used to solve many applied problems. However, mathematics gradually evolves towards a more abstract subject and is developed more independently from the applications. How all this came about becomes clear by reading this book.

The book is written as a textbook on the history of mathematics, and hence it is assumed that the reader has attended some analysis courses: real and preferably also complex analysis. There are also (few) end-of-chapter exercises which are clearly pointing to history students. That means that some topics are suggested to investigate and to report on them in an essay. There is also an end-of-course chapter giving advise of how to choose a topic for an essay and what kind of content it should be given (the mathematical technicalities are less important, as long as they are correct, but concentrate on why and how mathematical concepts grew into the ones that we know today).

The organization of the chapters is more or less chronological. One may recognize three parts: the first one sketches the situation in the early and the first half of the 19th century; the second part deals with the middle of the century, when complex analysis comes more to the foreground; and the third part is then announcing the transition to the 20th century, the foundations of mathematics are questioned, set theory, the real number system, topology all push mathematics into a more abstract framework.

Often the mathematician's findings were written down in books that grew out of lecture notes, which of course forced them to reflect upon the foundations of what they were teaching and these books were obviously very influential. The concepts of course still survive, but we would not always be satisfied with the way they were originally described.

The initial setting is made in the first three chapters with Lagrange, Fourier and Legendre. It is clear that these were interested in developing new ideas, not worrying too much about the basics or the finer details. Lagrange struggled with the foundations of analysis, his approach being basically algebraic without infinitesimals. Fourier had proposed his trigonometric series, and Legendre's contribution was to set up a theory of elliptic integrals. These elliptic functions form a recurrent topic in the next chapters as it was further developed by Abel who was the first to place them in the realm of complex functions, as did also Jacobi, Gauss, Liouville, and Hermite. In fact, Gray uses them as a kind of case study that extends over a large part of the book. Cauchy of course was influential on many other domains as well: continuity, series, differentiation and integration, and complex functions. He was the first to introduce more structure and rigor in real analysis. Equally productive was the master calculator Gauss with contributions on integration and complex analysis. These chapters span approximately the first half of the century. Gray interrupts here the development to give a reflection on what has been achieved so far.

The next half of the century starts with a new topic: potential theory (Green, Cauchy, Dirichlet). Riemann is given somewhat more attention in several chapters with his Riemann function, elliptic functions (the bread crumbs in this historical expedition), but of course also complex analysis and his conformal mapping theorem. There is also a discussion of how his work was received by his contemporaries. An alternative for Riemann's geometric function theory was provided by Weierstrass who initiated the concept of an analytic function. Here Gray reflects again on the past chapters. Still real analysis, and certainly complex analysis had not reached the rigor that we are used to.

This rigor was only starting to develop in what is the third part of this book. For example, it was only noted 20 years after its original formulation that Cauchy's theorem stating that the sum of (infinitely many) continuous functions was continuous did not always hold true. Only then, it was realized that functions could be much more exotic objects than the smooth curves that they were originally thought of. Mending Cauchy's theorem led to the concept of uniform convergence, non-differentiable and non-integrable functions (Bolzano, Cantor, Schwarz, Heine, Dini,...). This entailed Lebesgue's integration theory and the unavoidable rigorous definition of real number system, set theory, and topology.

It is also interesting to see what triggered the development of mathematics. Fourier used his series in heat diffusion problems, Legendre used elliptic integrals to study the mechanics of a pendulum, potential theory grew out of electromagnetic problems. However with rigor came also abstraction and, although still applicable, mathematics itself became the driving force for its own expansion.

The text is illustrated with portraits of the key mathematicians and where appropriate, plots to visualize some of the interesting functions or graphs are included. In several appendices, we find some translated papers (Fourier, Dirichlet, Riemann, Schwarz), and some more technical mathematical ones on series of functions and their convergence and on potential theory. Obviously a list of references (many of them being the original publications that were discussed, while others are more recent historical studies) as well as a mixed index of names and subjects are indispensable in a book like the present one.

It is very well known that history is of utmost importance and that we should learn from it. However, it is astonishing how fast even recent history is completely forgotten. This book certainly learns something to students but also to professional mathematicians, something that is too often neglected. Mathematics develops not by adding some epsilon improvements to an existing sequence of definitions, properties, theorems and proofs. The majority of the papers that are published are in that vein of thinking. With each one the boundaries of our mathematical knowledge crawls a bit forward. But what is actually progressing mathematics is the exploration of an unknown mine field. The explorers that venture there are the ones whose names will be printed in bold face in future history books. This book learns that these explorers of the past may have felt uncertain, made mistakes, and even used a trial and error approach. Only when the right track has been flagged, the road can be paved with the proper rigor needed to move on. So this book is not only an interesting read for the students who (have to) study it, but equally valuable for professional mathematicians. This is if they are prepared to take the time and reflect on the not so distant past of their beloved subject to which they want to contribute.

**Submitted by Adhemar Bultheel |

**9 / Feb / 2016