The first edition of this text was published 23 years ago and the second edition 11 years ago. It was stated in the Preface to the second edition that the tremendous development of high-speed computing devices was a major factor guiding the changes and revisions presented in that edition. The developments which have come about in digital computers since the publication of the second edition have been all the more spectacular. Computation speeds and memory size have increased by two orders of magnitude during this period, while machine costs have dropped by an order of magnitude. A single modern center, such as the Western Data Processing Center at the University of California at Los Angeles, has as much computing capacity today as all the combined installations in the United States a decade ago. There is also no apparent reason why one should not expect these trends to continue over the next decade.
Another factor to be considered is the recent development of time-sharing computer systems which, through large numbers of remote terminals, provide engineers and scientists with direct and immediate access to a computer for problem solving. It may not be too long until the remote terminal has replaced the slide rule as a readily available and sophisticated computing tool. As a result the analyst will be spending more time on developing realistic mathematical models for a physical problem and less time on the computational details. Also, the magnitude of the computations are of lesser concern because of the great speed of digital machines. However, computers have not eliminated the problem of choosing between accuracy and speed; they have only shifted the break-even point to a higher level.
We would like to suggest two particular areas of concern which we feel should be reviewed from time to time by every analyst. The first is to maintain an awareness of the limitations of any mathematical model resulting from the various approximations imposed during the modeling process. This is very important in order to avoid predicting the behavior of a system by a solution obtained from a model based on postulates which are invalid in the region of interest. Our second concern is that even though computers have given us the ability to study complex models, we should not stop seeking simpler representations, as it is an easy matter to overcomplicate a problem.
Perhaps the greatest factor creating the need for the extensive changes included in this third edition has been the widespread changes in engineering curricula which will continue for several years to come. Coupled with this are improvements in teaching mathematics at the high school level, which have enabled the shifting of more advanced material into the lower-division mathematics sequences in many colleges. These two situations have increased the mathematics requirements for engineering students and raised the level of mathematical rigor. The authors feel that both of these situations are desirable, but it was decided not to trade off the valuable physical applications for increased rigor in this edition. It is felt that most instructors teaching an advanced course based on this text can easily add any desired amount of rigor during the lectures but that it is usually more difficult to add a wide range of physical applications during the lecture period.
Those readers who are familiar with the previous two editions of this book will rapidly become aware of the extensive changes which have been made in this third edition. The arrangement of the chapters has been changed greatly, with those on series, special functions, vectors and tensors, transcendental equations, and partial differentiation being moved to appendixes. This change is an attempt to make the material more flexible for the variety of courses in which this book could be employed as a text. Since the material in the body of the text is designed for use in a one-year course, it is hoped that each instructor will feel free to add from, or exchange with, the material in the appendixes to meet the level and individual requirements of his course.
Other changes are the combining of the two previous tables of Laplace transforms into a single one in Appendix A, to which a few additional transform pairs have been added. The p-multiplied Laplace-transform notation has been dropped in favor of the more common s notation. Also additional problems have been added to each chapter, with answers and hints to selected problems supplied in Appendix G. Fourier, Hankel, and Mellin transforms have been added to the chapter on operational methods, and a new chapter on statistics and probability has been included.
As previously mentioned, this text is designed for use in a one-year course, and the chapters have been written to make them as independent as possible to give each instructor freedom in designing his own course. Chapters 1 to 8 are primarily concerned with the analysis of lumped parameter systems and could comprise the material for the first semester’s course with or without some combinations of material contained in the appendixes. Chapters 9 to 13 deal with distributed parameter systems, while Chapters 14 to 16 cover various important areas of applied mathematics. As such, these eight chapters could comprise the material for the second-semester’s course.
The authors would like to extend their appreciation to colleagues and students for their helpful assistance and insights in the preparation of this material. Thanks are also due to the editors of McGraw-Hill for their patience and assistance, and to the reviewers who contributed several helpful suggestions for improvement. Special thanks are due to Professor S. Takeda, Hosei University, Tokyo, for his detailed errata for the second edition which have been included in this present edition. Finally the greatest acknowledgment is due to our wives, Johanna and Doris, for their continued encouragement and understanding.
LOUIS A. PIPES
LAWRENCE R. HARVILL