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Nevanlinna's Theory of Value Distribution : The Second Main Theorem and its Error Terms - William Cherry

Nevanlinna's Theory of Value Distribution

The Second Main Theorem and its Error Terms

Hardcover Published: 24th April 2001
ISBN: 9783540664161
Number Of Pages: 203

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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.

Industry Reviews

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 Nevanlinna's 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, Zentralblatt MATH, Vol. 981, 2002) "The book is a very nice introduction to the theory of value distribution by covering all major aspects of the theory. ... One does not need too much prerequisite in reading the material, for the tools and concepts are introduced and explained in detail ... . This is a very valuable book that may be interesting both for researchers in function theory and number theory and for students." (Vilmos Totik, Acta Scientiarum Mathematicarum, Vol. 68, 2002)

A Word on Notation
Introductionp. 1
The First Main Theoremp. 5
The Poisson-Jensen Formulap. 5
The Nevanlinna Functionsp. 12
The First Main Theoremp. 17
Ramification and Wronskiansp. 20
Nevanlinna Functions for Sums and Productsp. 23
Nevanlinna Functions for Some Elementary Functionsp. 24
Growth Order and Maximum Modulusp. 27
A Number Theoretic Digression: The Product Formulap. 28
Some Differential Operators on the Planep. 31
Theorems of Stokes, Fubini, and Green-Jensenp. 33
The Geometric Interpretation of the Ahlfors-Shimizu Characteristicp. 35
Why N and T are Used in Nevanlinna Theory Instead of n and Ap. 41
Relationship Among the Nevanlinna Functions on Averagep. 43
Jensen's Inequalityp. 47
The Second Main Theorem via Negative Curvaturep. 49
Khinchin Functions and Exceptional Setsp. 49
The Nevanlinna Growth Lemma and the Height Transformp. 51
Definitions and Notationp. 56
The Ramification Theoremp. 57
The Second Main Theoremp. 60
A Simpler Error Termp. 71
The Unintegrated Second Main Theoremp. 73
A Uniform Second Main Theoremp. 74
The Spherical Isoperimetric Inequalityp. 78
Logarithmic Derivativesp. 89
Inequalities of Smirnov and Kolokolnikovp. 89
The Gol'dberg-Grinshtein Estimatep. 92
The Borel-Nevanlinna Growth Lemmap. 98
The Logarithmic Derivative Lemmap. 104
Functions of Finite Orderp. 106
The Second Main Theorem via Logarithmic Derivativesp. 107
Definitions and Notationp. 107
The Second Main Theoremp. 108
Functions of Finite Orderp. 117
Some Applicationsp. 119
Infinite Productsp. 120
Defect Relationsp. 123
Picard's Theoremp. 128
Totally Ramified Valuesp. 129
Meromorphic Solutions to Differential Equationsp. 129
Functions Sharing Valuesp. 135
Bounding Radii of Discsp. 136
Theorems of Landau and Schottky Typep. 141
Slowly Moving Targetsp. 144
Fixed Points and Iterationp. 146
A Further Digression into Number Theory: Theorems of Roth and Khinchinp. 151
Roth's Theorem and Vojta's Dictionaryp. 151
The Khinchin Convergence Conditionp. 158
More on the Error Termp. 161
Sharpness of the Second Main Theorem and the Logarithmic Derivative Lemmap. 161
Better Error Terms for Functions with Controlled Growthp. 170
Error Terms for Some Classical Special Functionsp. 177
Bibliographyp. 187
Glossary of Notationp. 193
Indexp. 197
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540664161
ISBN-10: 3540664165
Series: Springer Monographs in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 203
Published: 24th April 2001
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 1.08