Nevanlinnas Theory of Value Distribution: The Second Main Theorem and its Error [Hardcover]

On the one hand, this monograph serves as a self-contained introduction to Nevanlinna's theory of value distribution because the authors only assume the reader is familiar with the basics of complex analysis. On the other hand, the monograph also serves as a valuable reference for the research specialist because the authors present, for the first time in book form, the most modern and refined versions of the Second Main Theorem with precise error terms, in both the geometric and logarithmic derivative based approaches. A unique feature of the monograph is its number-theoretic digressions . These special sections assume no background in number theory and explore the exciting interconnections between Nevanlinna theory and the theory of Diophantine approximation.1. The First Main Theorem.- 2. The Second Main Theorem via Negative Curvature.- 3. Logarithmic Derivatives.- 4. The Second Main Theorem via Logarithmic Derivatives.- 5. Some Applications.- 6. A Further Digression into Number Theory: Theorems of Roth and Khinchin.- 7. More on the Error Term.- Glossary of Notation.

From the reviews of the first edition:

It is a phenomenon that value distribution theory is still a lively area of research. & As there is still need for further investigation which seems to require fundamentally new ideas, the authors were motivated to write this book. & The present book is very well written and it is a pleasure to read, especially the detailed motivations and comparisons of the different methods introduced and used in Nevanlinna theory. (Heinrich Begehr, Mathematical Reviews, Issue 2002 h)

This book collects together the existing work on error terms. It gives an introduction to Nevanlinnas theory of meromorphic functions, with an emphasis on error terms. & The book is very well-written. The reviewer highly recommends this book. To close on a personal note, I learned a lot from reading this book. (Min Ru, Zentralblatlã)