Analytic Solutions of Functional Equations.
Title:
Analytic Solutions of Functional Equations.
Author:
Cheng, Sui Sun.
ISBN:
9789812793355
Personal Author:
Physical Description:
1 online resource (296 pages)
Contents:
Contents -- Preface -- 1. Prologue -- 1.1 An Example -- 1.2 Basic Definitions -- 1.3 Notes -- 2. Sequences -- 2.1 Lebesgue Summable Sequences -- 2.2 Relatively Summable Sequences -- 2.3 Uniformly Summable Sequences -- 2.4 Properties of Univariate Sequences -- 2.4.1 Common Sequences -- 2.4.2 Convolution Products -- 2.4.3 Algebraic Derivatives and Integrals -- 2.4.4 Composition Products -- 2.5 Properties of Bivariate Sequences -- 2.6 Notes -- 3. Power Series Functions -- 3.1 Univariate Power Series Functions -- 3.2 Univariate Analytic Functions -- 3.3 Bivariate Power Series Functions -- 3.4 Bivariate Analytic Functions -- 3.5 Multivariate Power Series and Analytic Functions -- 3.6 Matrix Power Series and Analytic Functions -- 3.7 Majorants -- 3.8 Siegel's Lemma -- 3.9 Notes -- 4. Functional Equations without Differentiation -- 4.1 Introduction -- 4.2 Analytic Implicit Function Theorem -- 4.3 Polynomial and Rational Functional Equations -- 4.4 Linear Equations -- 4.4.1 Equation I -- 4.4.2 Equation II -- 4.4.3 Equation III -- 4.4.4 Equation IV -- 4.4.5 Equation V -- 4.4.6 Schr oder and Poincar e Equations -- 4.5 Nonlinear Equations -- 4.6 Notes -- 5. Functional Equations with Di erentiation -- 5.1 Introduction -- 5.2 Linear Systems -- 5.3 Neutral Systems -- 5.4 Nonlinear Equations -- 5.5 Cauchy-Kowalewski Existence Theorem -- 5.6 Functional Equations with First Order Derivatives -- 5.6.1 Equation I -- 5.6.2 Equation II -- 5.6.3 Equation III -- 5.6.4 Equation IV -- 5.6.5 Equation V -- 5.6.6 Equation VI -- 5.7 Functional Equations with Higher Order Derivatives -- 5.7.1 Equation I -- 5.7.2 Equation II -- 5.7.3 Equation III -- 5.7.4 Equation IV -- 5.8 Notes -- 6. Functional Equations with Iteration -- 6.1 Equations without Derivatives -- 6.1.1 Babbage Type Equations -- 6.1.2 Equations Involving Several Iterates -- 6.1.3 Equations of Invariant Curves.
6.1.3.1 Equation I -- 6.1.3.2 Equation II -- 6.2 Equations with First Order Derivatives -- 6.2.1 Equation I -- 6.2.2 Equation II -- 6.2.3 Equation III -- 6.2.4 Equation IV -- 6.2.5 First Order Neutral Equation -- 6.3 Equations with Second Order Derivatives -- 6.3.1 Equation I -- 6.3.2 Equation II -- 6.3.3 Equation III -- 6.3.4 Equation IV -- 6.4 Equations with Higher Order Derivatives -- 6.4.1 Equation I -- 6.4.2 Equation II -- 6.5 Notes -- Appendix A Univariate Sequences and Properties -- A.1 Common Sequences -- A.2 Sums and Products -- A.3 Quotients -- A.4 Algebraic Derivatives and Integrals -- A.5 Tranformations -- A.6 Limiting Operations -- A.7 Operational Rules -- A.8 Knowledge Base -- A.9 Analytic Functions -- A.10 Operations for Analytic Functions -- Bibliography -- Index.
Abstract:
This book presents a self-contained and unified introduction to the properties of analytic functions. Based on recent research results, it provides many examples of functional equations to show how analytic solutions can be found. Unlike in other books, analytic functions are treated here as those generated by sequences with positive radii of convergence. By developing operational means for handling sequences, functional equations can then be transformed into recurrence relations or difference equations in a straightforward manner. Their solutions can also be found either by qualitative means or by computation. The subsequent formal power series function can then be asserted as a true solution once convergence is established by various convergence tests and majorization techniques. Functional equations in this book may also be functional differential equations or iterative equations, which are different from the differential equations studied in standard textbooks since composition of known or unknown functions are involved. Sample Chapter(s). Chapter 1: Prologue (159 KB). Contents: Prologue; Sequences; Power Series Functions; Functional Equations without Differentiation; Functional Equations with Differentiation; Functional Equations with Iteration. Readership: Engineers, mathematicians, physicists, advanced undergraduates and graduates in mathematics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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