As a brilliant university lecturer, B. Ya. Levin attracted a large audience of working mathematicians and of students from various levels and backgrounds. For approximately 40 years, his Kharkov University seminar was a school for mathematicians working in analysis and a center for active research. This monograph aims to expose the main facts of the theory of entire functions and to give their applications in real and functional analysis. The general theory starts with the fundamental results on the growth of entire functions of finite order, their factorization according to the Hadamard theorem, properties of indicator and theorems of Phragmen-Lindelof type.

Mathematics is a beautiful subject, and entire functions is its most beautiful branch. Every aspect of mathematics enters into it, from analysis, algebra, and geometry all the way to differential equations and logic. For example, my favorite theorem in all of mathematics is a theorem of R. NevanJinna that two functions, meromorphic in the whole complex plane, that share five values must be identical. For real functions, there is nothing that even remotely corresponds to this. This book is an introduction to the theory of entire and meromorphic functions, with a heavy emphasis on Nevanlinna theory, otherwise known as value-distribution theory. Things included here that occur in no other book (that we are aware of) are the Fourier series method for entire and mero morphic functions, a study of integer valued entire functions, the Malliavin Rubel extension of Carlson's Theorem (the "sampling theorem"), and the first-order theory of the ring of all entire functions, and a final chapter on Tarski's "High School Algebra Problem," a topic from mathematical logic that connects with entire functions. This book grew out of a set of classroom notes for a course given at the University of Illinois in 1963, but they have been much changed, corrected, expanded, and updated, partially for a similar course at the same place in 1993. My thanks to the many students who prepared notes and have given corrections and comments.

This book is devoted to the theory of entire Hermitian operators, an important branch of functional analysis harmoniously combining the methods of operator theory and the theory of analytic functions. This theory anables various problems of classical and modern analysis to be looked at from a uniform point of view. In addition, it serves as a source for setting and solving many new problems in both theories. The three chapters of the book are based on the notes written by his students of M. G. Krein's lectures on the theory of entire operators with (1,1) deficiency index which he delivered in 1961 at the Pedagogical Institute of Odessa, and on his works on the extension theory of Hermitian operators and the theory of analytic functions. The theory is further developed in the direction of solving the problems set up by Krein at ICM-66 in the first two appendices. The first concerns the case of Hermitian operators with arbitrary defect numbers, entire with respect to an ordinary gauge and to a generalized one as well. The other focuses on the entire operators representable by differential operators. The third appendix is the translation from Russian of the unpublished notes of Krein's lecture in which, in particular, the place of the theory of entire operators in the whole analysis is elucidated. In Krein's mathematical heritage the theory of entire operators occupies a special position.

Entire Functions

In combinatorics, one often considers the process of enumerating objects of a certain nature, which results in a sequence of positive integers. With each such sequence, one can associate a generating function, whose properties tell us a lot about the nature of the objects being enumerated. Nowadays, the language of generating functions is the main language of enumerative combinatorics. This book is based on the course given by the author at the College of Mathematics of the Independent University of Moscow. It starts with definitions, simple properties, and numerous examples of generating functions. It then discusses various topics, such as formal grammars, generating functions in several variables, partitions and decompositions, and the exclusion-inclusion principle. In the final chapter, the author describes applications of generating functions to enumeration of trees, plane graphs, and graphs embedded in two-dimensional surfaces. Throughout the book, the reader is motivated by interesting examples rather than by general theories. It also contains a lot of exercises to help the reader master the material. Little beyond the standard calculus course is necessary to understand the book. It can serve as a text for a one-semester undergraduate course in combinatorics.

Fricke's groundbreaking study of the theory of elliptic modular functions is a must-read for anyone interested in the foundations of modern mathematics. With clear explanations and insightful examples, Fricke offers a comprehensive overview of this complex and fascinating subject. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

This book is devoted to the theory of entire Hermitian operators, an important branch of functional analysis harmoniously combining the methods of operator theory and the theory of analytic functions. This theory anables various problems of classical and modern analysis to be looked at from a uniform point of view. In addition, it serves as a source for setting and solving many new problems in both theories. The three chapters of the book are based on the notes written by his students of M. G. Krein's lectures on the theory of entire operators with (1,1) deficiency index which he delivered in 1961 at the Pedagogical Institute of Odessa, and on his works on the extension theory of Hermitian operators and the theory of analytic functions. The theory is further developed in the direction of solving the problems set up by Krein at ICM-66 in the first two appendices. The first concerns the case of Hermitian operators with arbitrary defect numbers, entire with respect to an ordinary gauge and to a generalized one as well. The other focuses on the entire operators representable by differential operators. The third appendix is the translation from Russian of the unpublished notes of Krein's lecture in which, in particular, the place of the theory of entire operators in the whole analysis is elucidated. In Krein's mathematical heritage the theory of entire operators occupies a special position.