Synopsis This book is an ideal text for an advanced course in the theory of complex functions. The author leads the reader to experience function theory personally and to participate in the work of the creative mathematician. The book contains numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. Topics covered include Weierstrass's product theorem, Mittag-Leffler's theorem, the Riemann mapping theorem, and Runge's theorems on approximation of analytic functions. In addition to these standard topics, the reader will find Eisenstein's proof of Euler's product formula for the sine function; Wielandt's uniqueness theorem for the gamma function; a detailed discussion of Stirling's formula; Iss'sa's theorem; Besse's proof that all domains in C are domains of holomorphy; Wedderburn's lemma and the ideal theory of rings of holomorphic functions; Estermann's proofs of the overconvergence theorem and Bloch's theorem; a holomorphic imbedding of the unit disc in C3; and Gauss's expert opinion of November 1851 on Riemann's dissertation.Remmert elegantly presents the material in short clear sections, with compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, and historical remarks, as well as the extensive bibliography, will make this book an invaluable source for students and teachers. From the Back Cover This book is an ideal text for an advanced course in the theory of complex functions. The author leads the reader to experience function theory personally and to participate in the work of the creative mathematician. The book contains numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. Topics covered include Weierstrass's product theorem, Mittag-Leffler's theorem, the Riemann mapping theorem, and Runge's theorems on approximation of analytic functions. In addition to these standard topics, the reader will find Eisenstein's proof of Euler's product formula for the sine function; Wielandt's uniqueness theorem for the gamma function and applications; a detailed discussion of Stirling's formula; Iss'sa's theorem; Besse's proof that all domains in C are domains of holomorphy; Wedderburn's lemma and the ideal theory of rings of holomorphic functions; Estermann's proofs of the overconvergence theorem and Bloch's theorem; a holomorphic imbedding of the unit disc in C(superscript 3); and Gauss's expert opinion of November 1851 on Riemann's dissertation. Remmert presents the material in short clear sections, with compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, and historical remarks, as well as the extensive bibliography, will make this book an invaluable source for students and teachers.
E**F
like the Jacobi triple product identity
This text by R. Remmert is lucid and quite interesting. It is very historical in nature, with work and letters by Gauss, Riemann, Jacobi and many others. A reader interested in classical function theory, a subject vivid two centuries ago, will find a gem in this book.Among the material is a expostion of the gamma and beta function and associated functions, some partition functions and related identities, like the Jacobi triple product identity, but also many more objects of that nature. ( This has relations to statistical and string physics, like D3-branes.). Also, the reader covers Issasas theorem-and I do not know any other book that does. Basically Issasas theorem says that a homomorphism between two meromorphic function C-algebras over two different domains( the precise meaning and setting is in the book) is given by a holomorphism mapping one domain to the other. The reader can imagine the applications :)!This is a good read, and I have recommended it to colleagues. Let us not forget what we learned two centuries ago!Mittag-Leffler and Weierstrass(The Mittag-Leffler theorem and a Weierstrass theorem on functions precribed by their poles and zeroes), Runge(Polynomial approximation of functions, etc), Montel/Ascoli-Arzela(Normal families of functions) and others are also in this book.You will also find the canonical Weierstrass product in this book, which actually also has applications in mathematical physics and partition functions. I'll give you an example of an application; If you use the Weierstrass product you will be able to derive the Dirac genus or A-roof genus in physics. Another application; you can use the Gamma function and it's behaviour, as expounded and described in this book, to compute physical amplitudes and other matters in Quantum Field Theory(QFT) via the process of dimensional renormalization, a procedure where you express physical amplitudes and other objects in terms of meromorphic functions of the dimension of the theory, basically curing pathological infinities that may arise by removing the purely meromorphic part of the germ, at the physical dimension, and keeping the holomorphic part.Remmerts book says nothing about physics, but it is good to have as a reference when you do physics, such as the above.So, this book has it's use!Thus, a great book by one of the great Germans!
B**O
Five Stars
This is one of the best mathematical books that I ever read!
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