Entire and Meromorphic Functions

✍ Scribed by Lee A. Rubel, James E. Colliander

• Publisher: Springer
• Year: 1996
• Tongue: English
• Leaves: 199
• Series: Universitext
• Edition: 1
• Category: Library

✦ Synopsis

This 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, and 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 closes with the Malliavin-Rubel uniqueness theorem. The approach 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.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
1. Introduction......Page 10
2. The Riemann-Stieltjes Integral......Page 12
3. Jensen's Theorem and Applications......Page 15
4. The First Fundamental Theorem of Nevanlinna Theory......Page 18
5. Elementary Properties of T(r, f)......Page 21
6. The Cartan Formulation of the Characteristic......Page 25
7. The Poisson-Jensen Formula......Page 29
8. Applications of T(r)......Page 32
9. A Lemma of Borel and Some Applications......Page 35
10. The Maximum Term of an Entire Function......Page 39
11. Relation Between the Growth of an Entire Function and the Size of Its Taylor Coefficients......Page 49
12. Carleman's Theorem......Page 54
13. A Fourier Series Method......Page 58
14. The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions......Page 87
15. Canonical Products......Page 96
16. Formal Power Series......Page 102
17. Picard's Theorem and the Second Fundamental Theorem......Page 108
18. A Proof of the Second Fundamental Theorem......Page 122
19. "Two Constant" Theorems and the Phragmen-Lindelof Theorems......Page 130
20. The Pblya Representation Theorem......Page 133
21. Integer-Valued Entire Functions......Page 148
22. On Small Entire Functions of Exponential-Type with Given Zeros......Page 155
23. The First-Order Theory of the Ring of All Entire Functions......Page 167
24. Identities of Exponential Functions......Page 184
References......Page 191
Index......Page 194
Back Cover......Page 199