Green's Functions and Infinite Products

Bridging the Divide

verfasst von: Yuri A. Melnikov

Verlag: Birkhäuser Boston

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook accounts for two seemingly unrelated mathematical topics drawn from two separate areas of mathematics that have no evident points of contiguity. Green's function is a topic in partial differential equations and covered in most standard texts, while infinite products are used in mathematical analysis. For the two-dimensional Laplace equation, Green's functions are conventionally constructed by either the method of images, conformal mapping, or the eigenfunction expansion. The present text focuses on the construction of Green's functions for a wide range of boundary-value problems.

Green's Functions and Infinite Products provides a thorough introduction to the classical subjects of the construction of Green's functions for the two-dimensional Laplace equation and the infinite product representation of elementary functions. Every chapter begins with a review guide, outlining the basic concepts covered. A set of carefully designed challenging exercises is available at the end of each chapter to provide the reader with the opportunity to explore the concepts in more detail. Hints, comments, and answers to most of those exercises can be found at the end of the text. In addition, several illustrative examples are offered at the end of most sections. This text is intended for an elective graduate course or seminar within the scope of either pure or applied mathematics.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Our objective in putting together this volume has been to develop a supplementary text for an elective upper-division undergraduate or graduate course/seminar that might be offered within the scope of a pure or applied mathematics curriculum. A quite unexpected treatment is delivered herein on two subjects that one might hardly have anticipated considering together in a single book. This makes the book an original and unique read, and a good choice for those who are open to challenges and welcome the unexpected.

Yuri A. Melnikov

Chapter 2. Infinite Products and Elementary Functions

Abstract

The objective in this chapter is to lay out a working background for dealing with infinite products and their possible applications. The reader will be familiarized with a specific topic that is not often included in traditional texts on related courses of mathematical analysis, namely the infinite product representation of elementary functions.

Yuri A. Melnikov

Chapter 3. Green’s Functions for the Laplace Equation

Abstract

Our recent work reported in Melnikov (Appl. Math. Sci. 2 (2008) 81–97 and J. Math. Anal. Appl. 344 (2008) 521–534) provides convincing evidence of a surprising linkage between the topics of approximation of functions and the Green’s function for some partial differential equations. The linkage appears promising and extremely productive. It has generated an unlooked-for approach to the infinite product representation of elementary functions.

Yuri A. Melnikov

Chapter 4. Green’s Functions for ODE

Abstract

As was convincingly shown in Chap. 3, the methods of images and conformal mapping are helpful in obtaining Green’s functions for the two-dimensional Laplace equation. But it is worth noting, at the same time, that the number of problems for which these methods are productive, is notably limited. To support this assertion, recall that mixed boundary-value problems with Robin conditions imposed on a piece of the boundary are not within the reach of these methods.

Yuri A. Melnikov

Chapter 5. Eigenfunction Expansion

Abstract

Having departed for a while from the main focus of the book in the previous chapter, where the emphasis was on ordinary differential equations, we are going to return in the present chapter to partial differential equations. The reader will be provided with a comprehensive review of another approach that has been traditionally employed for the construction of Green’s functions for partial differential equations. The method of eigenfunction expansion will be used, representing one of the most productive and recommended methods in the field.

Yuri A. Melnikov

Chapter 6. Representation of Elementary Functions

Abstract

While the first five chapters in this book have touched upon more or less standard topics, the material of the present chapter goes in another direction. The reader will probably find it surprising. Indeed, the notions of infinite product and Green’s function, discussed in detail earlier in this volume, have customarily been included in texts on mathematical analysis and differential equations, respectively. The present chapter, in contrast, discusses an unusual idea that has never been explored in texts before. That is, a technique, reported for the first time in Melnikov (Appl. Math. Sci. 2 (2008) 81–97 and J. Math. Anal. Appl. 344 (2008) 521–534), is employed here for obtaining infinite product representations for a number of elementary functions.

Yuri A. Melnikov

Chapter 7. Hints and Answers to Chapter Exercises

Abstract

Keeping in mind that one of the profound ways to foster conceptual understanding is problem solving and trying to assist the reader with a better comprehension of the book material, we present below hints and answers to most chapter exercises.

Yuri A. Melnikov

Backmatter

Titel: Green's Functions and Infinite Products
verfasst von: Yuri A. Melnikov
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-8280-4
Print ISBN: 978-0-8176-8279-8
DOI: https://doi.org/10.1007/978-0-8176-8280-4