Green's Functions and Infinite Products: Bridging the Divide

ISBN-13: 9780817682798 / Angielski / Twarda / 2011 / 165 str.

Yuri A Melnikov

Opis

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.

Recenzje

From the reviews:

"The book under review is an interesting textbook which is intended for students (both undergraduate and postgraduate) specializing in pure and applied mathematics. It may be considered as a good complement to standard courses in analysis and differential equations." (Konstantin Yu. Fedorovski , Mathematical Reviews, May, 2013)

"The present book provides an introduction to some recent research of the author based on a method for deriving infinite product representations for the Green's functions of the two-dimensional Laplace equation on certain domains. ... The book contains ample background material on infinite products and Green's functions ... and is, hence, accessible for graduate students. In particular, the presentation is clear and self-contained." (G. Teschl, Monatshefte für Mathematik, Vol. 166 (3-4), June, 2012)

"This book is devoted to some new infinite products for trigonometric and hyperbolic functions. ... the infinite products investigated in the book have the same speed of convergence as the classical ones. ... The book is written on a very comprehensive level and can be useful for students studying equations of mathematical physics and computer modeling." (Yana Kinderknecht, Zentralblatt MATH, Vol. 1235, 2012)

Spis treści

INTRODUCTION.- CHAPTER 1: Infinite Products & Elementary Functions.- 1.1 Classical Euler representations.- 1.2 Alternative derivations.- 1.3 Other elementary functions.- 1.4 Chapter exercises.- CHAPTER 2: Green's Functions for the Laplace Equation.- 2.1 Construction by the method of images.- 2.2 Conformal mapping method.- 2.3 Chapter exercises.- CHAPTER 3: Green's Functions for ODE.- 3.1 Construction by defining properties.- 3.2 Method of variation of parameters.- 3.3 Chapter exercises.- CHAPTER 4: Method of Eigenfunction Expansion.- 4.1 Hilbert's theorem.- 4.2 Cartesian coordinates.- 4.3 Polar coordinates.- 4.4 Chapter exercises.- CHAPTER 5: New Infinite Product Representations.- 5.1 Method of images extends frontiers.- 5.2 Trigonometric functions.- 5.3 Hyperbolic functions.- 5.4 Chapter exercises.- HINTS AND ANSWERS TO CHAPTER EXERCISES.- REFERENCES.- INDEX.