Entire and meromorphic functions

This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the Écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field The book is an introduction to the theory of entire and meromorphic functions intended for advanced graduate students in mathematics and for professional mathematicians. The book provides a clear treatment of the Nevanlinna theory of value distribution of meromorphic functions, starting from scratch. It contains the first book-form presentation of the Rubel-Taylor Fourier series method for meromorphic functions and the Miles theorem on efficient quotient representation. It has a concise but complete treatment of the Polya theory of the Borel transform and the conjugate indicator diagram. It contains some of Buck's results on integer-valued entire functions, and the Malliavin-Rubel uniqueness theorem. The book closes with applications to mathematical logic.In particular, the first-order theory of the ring of entire functions is developed and questions concerning identities of exponential functions are studied as in Tarski's "High School Algebra Problems." The approach of the book gets to the heart of the matter without excessive scholarly detours. It prepares the reader for further study of the vast literature on the subject, which is one of the cornerstones of complex analysis. 1. Introduction -- 2. The Riemann-Stieltjes Integral -- 3. Jensen's Theorem and Applications -- 4. The First Fundamental Theorem of Nevanlinna Theory -- 5. Elementary Properties of T(r, f) -- 6. The Cartan Formulation of the Characteristic -- 7. The Poisson-Jensen Formula -- 8. Applications of T(r) -- 9. A Lemma of Borel and Some Applications -- 10. The Maximum Term of an Entire Function -- 11. Relation Between the Growth of an Entire Function and the Size of Its Taylor Coefficients -- 12. Carleman's Theorem -- 13. A Fourier Series Method -- 14. The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions -- 15. Canonical Products -- 16. Formal Power Series -- 17. Picard's Theorem and the Second Fundamental Theorem -- 18. A Proof of the Second Fundamental Theorem -- 19. "Two Constant" Theorems and the Phragmen-Lindelof Theorems -- 20. The Polya Representation Theorem -- 21. Integer-Valued Entire Functions