Entire Functions
Entire Functions
Published on 29. August 2011
275 pages
978-0-08-087313-8 (ISBN)
Description
Entire Functions
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Content
- Front Cover
- Entire Functions
- Copyright Page
- Contents
- Preface
- Chapter 1. Introduction
- 1.1. Terminology
- 1.2. Jensen's, Carleman's, and Nevanlinna's formulas
- 1.3. Carathéodory's inequality
- 1.4. Phragmén-Lindelöf theorems
- 1.5. Density of sequences and sets
- 1.6. Stirling's formula.
- 1.7. Mittag-Leffler summability
- 1.8. Laplace and Fourier transforms
- Chapter 2. General Properties of Entire Functions of Finite Order
- 2.1. Measures of rate of growth
- 2.2. Order and type in terms of the coefficients
- 2.3. Other connections between growth and coefficients
- 2.4. The order and type of a derivative
- 2.5. Rate of growth and distribution of zeros
- 2.6. Lemmas on infinite products
- 2.7. Hadamard's factorization theorem
- 2.8. Laguerre's theorem on separation of zeros
- 2.9. The zeros of functions of nonintegral order
- 2.10. The zeros of functions of integral order
- 2.11. Further relations between growth and zeros
- 2.12. Functions of genus 0
- Chapter 3. The Minimum Modulus
- 3.1. Functions of order less than ½
- 3.2. Functions of order less than 1
- 3.3. Functions of order 1
- 3.4. The minimum modulus of a polynomial
- 3.5. Lemmas on functions of small order
- 3.6. Functions of order zero
- 3.7. Functions of larger order
- Chapter 4. Functions with Real Negative Zeros
- 4.1. Direct theorems
- 4.2. Converse theorems
- 4.3. Generalizations
- 4.4. Another kind of theorem
- Chapter 5. General Properties of Functions of Exponential
- 5.1. Properties of the indicator
- 5.2. Convex sets
- 5.3. The indicator diagram
- 5.4. Properties of the indicator diagram
- 5.5. The Borel transform on the boundary of the conjugate indicator diagram
- 5.6. Functions of exponential type in an angle
- Chapter 6. Functions of Exponential Type. Restricted on a Line. I. Theormes in the Large
- 6.1. Introduction
- 6.2. Functions bounded on a line
- 6.3. Relations among integrals, mean values, and distribution of zeros
- 6.4. Generalizations of boundedness
- 6.5. Representation of functions in a half plane
- 6.6. Consequences of the representation theorem
- 6.7. Integrability on a line
- 6.8. Representations for functions which are bounded on a line
- 6.9. The indicator diagram of a finite Fourier transform
- 6.10. Periodic and almost periodic functions
- Chapter 7. Functions of Exponential Type, Restricted on a Line. II. Asymptotic Behavior in a Half Plane
- 7.1. Introduction
- 7.2. The asymptotic behavior of a Blaschke product
- 7.3. Asymptotic behavior with other boundary conditions
- 7.4. The limit of r-1 log M ( r )
- 7.5. Factorization of a positive function
- 7.6. The type of a product
- 7.8. Functions with a zero-free half plane
- Chapter 8. Functions of Exponential Type : Connections Between Growth and Distribution of Zeros
- 8.1. Introduction
- 8.2. Entire functions with real zeros
- 8.3. Entire functions with real zeros, continued
- 8.4. Entire functions with zeros close to the real axis
- Chapter 9. Uniqueness Theorems
- 9.1. Introduction
- 9.2. Zeros at the positive integers
- 9.3. Applications of Carleman's theorem
- 9.4. Zeros at the positive and negative integers
- 9.5. General distribution of zeros on a half line
- 9.6. Zeros near the integers
- 9.7. Zeros less regularly distributed
- 9.8. Functions with a sequence of small values
- 9.9. Alternation theorems
- 9.10. Uniqueness theorems with operators
- 9.11. Generalized Abel series
- 9.12. Integral-valued entire functions
- Chapter 10. Growth Theorems
- 10.1. Exponential growth on an arithmetic progression
- 10.2. Boundcdness on an arithmetic progression
- 10.3. Exponential growth on a sequence having a density
- 10.4. An application
- 10.5. Boundedness on a sequence which is close to an arithmetic progression
- 10.6. Applications to Lp and lp problems
- 10.7. Generalizations
- Chapter 11. Operators and Their Extremal Properties
- 11.1. Introduction
- 11.2. A general method
- 11.3. Bounds for derivatives
- 11.4. Inequalities involving the maximum of I f(x) I
- 11.5. Inequalities involving sup I f (n p / r) I
- 11.6. Extension of operators
- 11.7. A class of operators
- Chapter 12. Applications
- 12.1. Introduction
- 12.2. Asymptotic behavior of a mean value
- 12.3. A theorem on convolutions
- 12.4. Completeness of sets of functions
- 12.5. Fourier series
- 12.6. Power series on the circle of convergence
- 12.7. Dirichlet series
- 12.8. Gap theorems for entire functions
- 12.9. Expansions of analytic functions in series of polynomials
- 12.10. Differential equations of infinite order
- 12.11. Approximation by entire functions
- Bibliography
- Index
- Index of Notations
