Entire and meromorphic functions
The 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, starting from scratch. It contains the first book-form presentation of the Rubel-Taylor Fourier series method for meromorphic functions and the Miles theorem on efficient quotient representation. It has a concise but com…
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- Full title :Entire and meromorphic functions / Lee A. Rubel with assistance from James E. Colliander.
Summary
The 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, starting from scratch. It contains the first book-form 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 the Malliavin-Rubel uniqueness theorem. The book closes with applications to mathematical logic.
In particular, the first-order theory of the ring of entire functions is developed and questions concerning identities of exponential functions are studied as in Tarski's "High School Algebra Problems." The approach of the book 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.
Edition Statements
- Publication Date:©1996.
Publication Statements
- Publisher(s):Springer
- Place of Publication:New York
- Language:eng
- Publication Date:©1996.
The Physical Object
- Description:viii, 187 pages : : illustrations ; : 24 cm.
Edition Identifiers
- ISBN:0387945105, 9780387945101
- oclc:ocm33334260
- LC Classification:QA353.E5
- Dewey Decimal:515/.98
Table of Contents
1. Introduction -- 2. The Riemann-Stieltjes Integral -- 3. Jensen's Theorem and Applications -- 4. The First Fundamental Theorem of Nevanlinna Theory -- 5. Elementary Properties of T(r, f) -- 6. The Cartan Formulation of the Characteristic -- 7. The Poisson-Jensen Formula -- 8. Applications of T(r) -- 9. A Lemma of Borel and Some Applications -- 10. The Maximum Term of an Entire Function -- 11. Relation Between the Growth of an Entire Function and the Size of Its Taylor Coefficients -- 12. Carleman's Theorem -- 13. A Fourier Series Method -- 14. The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions -- 15. Canonical Products -- 16. Formal Power Series -- 17. Picard's Theorem and the Second Fundamental Theorem -- 18. A Proof of the Second Fundamental Theorem -- 19. "Two Constant" Theorems and the Phragmen-Lindelof Theorems -- 20. The Polya Representation Theorem -- 21. Integer-Valued Entire Functions.
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