Entire Functions
Description
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 135 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 a student 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.
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Persons
Gary Gundersen is a Research Professor Emeritus of Mathematics from the University of New Orleans (UNO), where he was a faculty member for thirty-one years. He has fifty research papers that are mainly in complex analysis and complex differential equations. He obtained his PhD in Mathematics from Rutgers University in 1975 when he then became a faculty member at UNO. He has given numerous talks at meetings and universities and he was the plenary speaker at two meetings. Gundersen was awarded a University Research Professorship at UNO in 1999. He was the main organizer of a special session at the 2001 Annual AMS meeting in New Orleans. At UNO he taught a variety of mathematics courses including a course on entire and meromorphic functions.
Janne Heittokangas is a Lecturer from the University of Eastern Finland. He has supervised 10 PhD students, 25 MSc students and 60 BSc students, and he received a good teacher award in 2013 and in 2025. Heittokangas has 60 research papers in analytic and meromorphic functions and in complex differential and difference equations. He obtained his MSc degree in 1996 and his PhD degree in 2000, both from the University of Joensuu. From 2002 to 2005 he worked as a J.L. Doob Research Assistant Professor at the University of Illinois at Urbana-Champaign. Heittokangas has served as an editor and a reviewer for multiple scientific journals, and he has been a member of the organizing committee for several scientific meetings and conferences.