调和函数理论 第2版（影印版）

　　 Harmonic functions--the solutions of Laplace's equation--play a crucial role in many areas of mathematics, physics, and engineering. But learning about them is not always easy. At times the authors have agreed with Lord Kelvin and Peter Tait, who wrote ([18], Preface)
　　 There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students.
　　

preface
acknowledgments
chapter 1
basic properties of harmonic functions
definitions and examples
invariance properties
the mean-value property
the maximum principle
the poisson kernel for the ball
the dirichlet problem for the ball
converse of the mean-value property
real analyticity and homogeneous expansions
origin of the term "harmonic"
exercises
chapter 2
bounded harmonic functions
liouvfile's theorem
isolated singularities
cauchy's estimates
normal families

　　 Harmonic functions--the solutions of Laplace's equation--play a crucial role in many areas of mathematics, physics, and engineering. But learning about them is not always easy. At times the authors have agreed with Lord Kelvin and Peter Tait, who wrote ([18], Preface)
　　 There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students.
　　The quotation has been included mostly for the sake of amusement, but it does convey a sense of the difficulties the uninitiated sometimes encounter.
　　 The main purpose of our text, then, is to make learning about harmonic functions easier. We start at the beginning of the subject, assuming only that our readers have a good foundation in real and complex analysis along with a knowledge of some basic results from functional analysis. The first fifteen chapters of [15], for example, provide sufficient preparation.
　　 In several cases we simplify standard proofs. For example, we replace the usual tedious calculations showing that the Kelvin transform of a harmonic function is harmonic with some straightforward observations that we believe are more revealing. Another example is our proof of B6cher's Theorem, which is more elementary than the classical proofs.
　　 We also present material not usually covered in standard treatments of harmonic functions (such as [9], [11], and [19]). The section on the Schwarz Lemma and the chapter on Bergman spaces are examples. For completeness, we include some topics in analysis that frequently slip through the cracks in a beginning graduate student's curriculum, such as real-analytic functions.
　　 We rarely attempt to trace the history of the ideas presented in this book. Thus the absence of a reference does not imply originality on our part.
　　 For this second edition we have made several major changes. The key improvement is a new and considerably simplified treatment of spherical harmonics (Chapter 5). The book now includes a formula for the Laplacian of the Kelvin transform (Proposition 4.6). Another addition is the proof that the Dirichlet problem for the half-space with continuous boundary data is solvable (Theorem 7.11), with no growth conditions required for the boundary function. Yet another significant change is the inclusion of generalized versions of Liouville's and B6cher's Theorems (Theorems 9.10 and 9.11), which are shown to be equivalent. We have also added many exercises and made numerous small improvements.
　　 In addition to writing the text, the authors have developed a software package to manipulate many of the expressions that arise in harmonic function theory. Our software package, which uses many results from this book, can perform symbolic calculations that would take a prohibitive amount of time if done without a computer. For example, the Poisson integral of any polynomial can be computed exactly. Appendix B explains how readers can obtain our software package free of charge.
　　 The roots of this book lie in a graduate course at Michigan State University taught by one of the authors and attended by the other authors along with a number of graduate students. The topic of harmonic functions was presented with the intention of moving on to different material after introducing the basic concepts. We did not move on to different material. Instead, we began to ask natural questions about harmonic functions. Lively and illuminating discussions ensued. A freewheeling approach to the course developed; answers to questions someone had raised in class or in the hallway were worked out and then presented in class (or in the hallway). Discovering mathematics in this way was a thoroughly enjoyable experience. We will consider this book a success if some of that enjoyment shines through in these pages.