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Localization and perturbation of zeros of entire functions. (English) Zbl 1195.30003

Lecture Notes in Pure and Applied Mathematics 258. Boca Raton, FL: CRC Press (ISBN 978-1-4398-0032-4/hbk; 978-1-138-11678-8/pbk). xvi, 300 p. (2010).
A major topic in the theory of entire functions is to relate the distribution of zeros of an entire function to the growth, the Taylor coefficients, and other properties of the function. Most of the results of this type are of an asymptotic nature. In contrast, this book is concerned with bounds for the zeros \(z_k(f)\) of an entire function \(f\) which are valid for all \(k\).
In particular, after four chapters with introductory and background material, the sums
\[ \sum_{k=1}^j \frac{1}{|z_k(f)|}\qquad\text{and}\qquad \sum_{k=1}^\infty \frac{1}{|z_k(f)|^p}, \] where \(p\) is greater than or equal to the order of \(f\), are studied in detail in Chapter 5. Chapter 6 addresses the question how close the zeros of two entire functions are if their Taylor coefficients are close. Chapters 7–9 are devoted to special classes of entire functions: functions of order less than two, functions of exponential type, and quasipolynomials (which are defined as finite linear combinations of functions of the form \(z\mapsto z^m e^{\lambda z}\)). Chapter 10 is concerned with generalized Borel transforms and estimates of canonical products. Finally, the last two chapters deal with matrix-valued polynomials and entire functions.
A considerable part of the results described in this book is due to the author.

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30D20 Entire functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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