Entire Functions

Entire functions are everywhere in mathematics, and they also occur frequently in other fields such as engineering and physics. They appear so naturally that it is often not even stated that they are entire functions. Starting with the growth and zeros of entire functions, this book then moves on to further classical topics such as the Phragmén-Lindelöf Principle, Wiman-Valiron theory, and Nevanlinna theory. Detailed discussions are given on special functions, which include entire functions with real zeros, exponential polynomials, and entire functions that have a particular asymptotic growth or certain regular growth conditions. Three chapters are devoted to entire function solutions of complex differential equations, which has been an active area of research for the last forty-five years. Last, there are sections on modern measures of growth, asymptotic values, Bank-Laine functions and other topics. As a reader-friendly introduction to entire functions, the book contains both the classical theory plus many results from the last sixty years that are not contained in older books. The proofs are step-by-step with details that sometimes fill in gaps and explain confusing discussions that are found in some original papers and older books. The book can be used in different ways. Chapters 1-5 contain basic material on entire functions that can be used by an instructor to teach a class to students that have a background in complex analysis. With this in mind, these five chapters purposely contain 137 practice problems of varying difficulty to help students learn the subject more thoroughly. Chapters 6-12 contain more recent results and provide insight into modern research. After students have taken a class on Chapters 1-5, an instructor could conduct a seminar for the students on some or all of the topics in Chapters 6-12, or students could read on their own about topics that they are curious about. Researchers can use the book as a good reference source or to refresh their knowledge on various topics.