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Entire function

About: Entire function is a research topic. Over the lifetime, 5081 publications have been published within this topic receiving 59410 citations. The topic is also known as: integral function.


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MonographDOI
31 Dec 1964

1,788 citations

Book
31 Dec 1934
TL;DR: In this article, a generalized harmonic analysis in the complex domain of random functions has been proposed, based on Szasz's theorem and a class of singular integral equations of the exponential type.
Abstract: Introduction Quasi-analytic functions Szasz's theorem Certain integral expansions A class of singular integral equations Entire functions of the exponential type The closure of sets of complex exponential functions Non-harmonic Fourier series and a gap theorem Generalized harmonic analysis in the complex domain The harmonic analysis of random functions Bibliography Index.

1,416 citations

Book
01 Jan 1960
TL;DR: In this paper, Cauchy-Goursat Theorem 2.1.1 was extended to include the point at infinity and the point of infinity at infinity in the definition of differentiability of analytical functions.
Abstract: 1 Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane 2 Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle 3 Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 4 Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Some Examples Examples with Branch Cuts Upper Bounds for Moduli of Contour Integrals Antiderivatives Proof of the Theorem Cauchy-Goursat Theorem Proof of the Theorem Simply Connected Domains Multiply Connected Domains Cauchy Integral Formula An Extension of the Cauchy Integral Formula Some Consequences of the Extension Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series Convergence of Sequences Convergence of Series Taylor Series Proof of Taylor's Theorem Examples Laurent Series Proof of Laurent's Theorem Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series 6 Residues and Poles Isolated Singular Points Residues Cauchy's Residue Theorem Residue at Infinity The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Inverse Laplace Transforms Examples 8 Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions 9 Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions 10 Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder 11 The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz Integral Formula Dirichlet Problem for a Half Plane Neumann Problems Appendixes Bibliography Table of Transformations of Regions Index

1,167 citations

Book
01 Jun 1996
TL;DR: In this article, the authors considered the problem of the growth of an entire function and the distribution of its zeros, and they gave a lower bound on the maximum number of zeros of a function in the half-plane.
Abstract: Part I. Entire Functions of Finite Order: Growth of entire functions Main integral formulas for functions analytic in a disk Some applications of the Jensen formula Factorization of entire functions of finite order The connection between the growth of an entire function and the distribution of its zeros Theorems of Phragmen and Lindelof Subharmonic functions The indicator function The Polya Theorem Applications of the Polya Theorem Lower bounds for analytic and subharmonic functions Entire functions with zeros on a ray Entire functions with zeros on a ray (continuation) Part II. Entire Functions of Exponential Type: Integral representation of functions analytic in the half-plane The Hayman Theorem Functions of class $C$ and their applications Zeros of functions of class $C$ Completeness and minimality of system of exponential functions in $L^2(0,a)$ Interpolation by entire functions of exponential type Interpolation by entire functions of the spaces $L_\pi$ and $B_\pi$ Sin-type functions Riesz bases formed by exponential functions in $L^2(-\pi,\pi)$ Completeness of the eigenfunction system of a quadratic operator pencil Part III. Some Additional Problems of the Theory of Entire Functions: Carleman's and R. Nevanlinna's formulas and their applications Uniqueness problems for Fourier transforms and for infinitely-differentiable functions The Matsaev Theorem on the growth of entire functions admitting a lower bound Entire functions of the class $P$ S. N. Bernstein's inequality for entire functions of exponential type and its generalizations Bibliography Author index Subject index.

941 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202322
202264
2021196
2020185
2019174
2018180