An Introduction to Laplace Transforms and Fourier Series (Springer Undergraduate Mathematics Series)

This book has been primarily written for the student of mathematics who is in the second year or the early part of the third year of an undergraduate course. It will also be very useful for students of engineering and the physical sciences for whom Laplace Transforms continue to be an extremely useful tool. The book demands no more than an elementary knowledge of calculus and linear algebra of the type found in many first year mathematics modules for applied subjects. For mathematics majors and specialists, it is not the mathematics that will be challenging but the applications to the real world. The author is in the privileged position of having spent ten or so years outside mathematics in an engineering environment where the Laplace Transform is used in anger to solve real problems, as well as spending rather more years within mathematics where accuracy and logic are of primary importance. This book is written unashamedly from the point of view of the applied mathematician. The Laplace Transform has a rather strange place in mathematics. There is no doubt that it is a topic worthy of study by applied mathematicians who have one eye on the wealth of applications; indeed it is often called Operational Calculus. Cover......Page 1 Springer Undergraduate Mathematics Series......Page 2 Title ......Page 4 ISBN 1-85233-015-5......Page 5 Dedication......Page 6 Preface ......Page 8 Contents ......Page 12 1.1 Introduction......Page 14 1.2 The Laplace Transform......Page 15 1.3 Elementary Properties......Page 18 1.4 Exercises......Page 24 2.1 Real Functions......Page 26 2.2 Derivative Property of the Laplace Transform......Page 27 2.3 Heaviside's Unit Step Function......Page 31 2.4 Inverse Laplace Transform......Page 32 2.5 Limiting Theorems......Page 36 2.6 The Impulse Function......Page 38 2. 7 Periodic Functions......Page 45 2.8 Exercises......Page 47 3.2 Convolution......Page 50 3.3 Ordinary Differential Equations......Page 62 3.3.1 Second Order Differential Equations......Page 67 3.3.2 Simultaneous Differential Equations......Page 76 3.4 Using Step and Impulse Functions......Page 81 3.5 Integral Equations......Page 86 3 . 6 Exercises......Page 88 4.1 lntrod uction......Page 92 4.2 Definition of a Fourier Series......Page 94 4.3 Odd and Even Functions......Page 104 4.4 Complex Fourier Series......Page 107 4.6 Properties of Fourier Series......Page 114 4.7 Exercises......Page 121 5.1 Introduction......Page 124 5.2 Classification of Partial Differential Equations......Page 126 5.3 Separation of Variables......Page 128 5.4 Using Laplace Transforms to Solve PDEs......Page 131 5.5 Boundary Conditions and Asymptotics......Page 136 5.6 Exercises......Page 139 6.2 Deriving the Fourier Trans......Page 142 6.3 Basic Properties of the Fourier Transform......Page 147 6.4 Fo urier Transforms and Partial Differential Equations......Page 155 6.5 Signal Processing......Page 159 6.6 Exercises......Page 166 7.2 Rudiments of Complex Analysis......Page 170 7.3 Complex Integration......Page 173 7.4 Branch Points......Page 180 7.5 The Inverse Laplace Transform......Page 185 7.6 Using the Inversion Formula in Asymptotics......Page 190 7.7 Exercises......Page 194 1, 2......Page 198 3, 4, 5......Page 199 6, 7, 8......Page 200 1, 2, 3......Page 201 4, 5, 6, 7......Page 202 8, 9, 10......Page 203 11, 12......Page 204 13, 14......Page 205 1......Page 206 2, 3......Page 207 4, 5......Page 208 6......Page 210 7, 8, 9, 10......Page 212 1......Page 213 2, 3, 4......Page 214 5, 6, 7......Page 215 8.......Page 216 9.......Page 217 10, 11......Page 218 12......Page 219 13......Page 220 14.......Page 221 1......Page 222 2, 3, 4......Page 223 5, 6......Page 224 7, 8......Page 225 1......Page 226 2, 3, 4......Page 227 5, 6, 7......Page 228 8......Page 229 9, 10, 11......Page 230 1, 2......Page 231 3......Page 232 4......Page 233 5.......Page 234 6......Page 236 7, 8......Page 237 9, 10......Page 238 B. Table of Laplace Transforms ......Page 240 C,1 Linear Algebra......Page 244 C.2 Gramm-Schmidt Orthonormalisation Process......Page 256 Bibliography ......Page 258 Index ......Page 260 Cover 1 Springer Undergraduate Mathematics Series 2 Title 4 Copyright 5 © Springer-Verlag London Linlited 2001 5 ISBN 1-85233-015-5 5 Dedication 6 Preface 8 Contents 12 I. The Laplace Transform 14 1.1 Introduction 14 1.2 The Laplace Transform 15 1.3 Elementary Properties 18 1.4 Exercises 24 II. Further Properties of the Laplace Transform 26 2.1 Real Functions 26 2.2 Derivative Property of the Laplace Transform 27 2.3 Heaviside's Unit Step Function 31 2.4 Inverse Laplace Transform 32 2.5 Limiting Theorems 36 2.6 The Impulse Function 38 2. 7 Periodic Functions 45 2.8 Exercises 47 III. Convolution and the Solution of Ordinary Differential Equations 50 3.1 Introduction 50 3.2 Convolution 50 3.3 Ordinary Differential Equations 62 3.3.1 Second Order Differential Equations 67 3.3.2 Simultaneous Differential Equations 76 3.4 Using Step and Impulse Functions 81 3.5 Integral Equations 86 3 . 6 Exercises 88 IV. Fourier Series 92 4.1 lntrod uction 92 4.2 Definition of a Fourier Series 94 4.3 Odd and Even Functions 104 4.4 Complex Fourier Series 107 4.6 Properties of Fourier Series 114 4.7 Exercises 121 V. Partial Differential Equations 124 5.1 Introduction 124 5.2 Classification of Partial Differential Equations 126 5.3 Separation of Variables 128 5.4 Using Laplace Transforms to Solve PDEs 131 5.5 Boundary Conditions and Asymptotics 136 5.6 Exercises 139 VI. Fourier Transforms 142 6.1 Introduction 142 6.2 Deriving the Fourier Trans 142 6.3 Basic Properties of the Fourier Transform 147 6.4 Fo urier Transforms and Partial Differential Equations 155 6.5 Signal Processing 159 6.6 Exercises 166 VII. Complex Variables and Laplace Transforms 170 7.1 Introduction 170 7.2 Rudiments of Complex Analysis 170 7.3 Complex Integration 173 7.4 Branch Points 180 7.5 The Inverse Laplace Transform 185 7.6 Using the Inversion Formula in Asymptotics 190 7.7 Exercises 194 A. Solutions to Exercises 198 Exercises 1.4 198 1, 2 198 3, 4, 5 199 6, 7, 8 200 Exercises 2.8 201 1, 2, 3 201 4, 5, 6, 7 202 8, 9, 10 203 11, 12 204 13, 14 205 Exercises 3.6 206 1 206 2, 3 207 4, 5 208 6 210 7, 8, 9, 10 212 Exercises 4.7 213 1 213 2, 3, 4 214 5, 6, 7 215 8. 216 9. 217 10, 11 218 12 219 13 220 14. 221 Exercises 5.6 222 1 222 2, 3, 4 223 5, 6 224 7, 8 225 9 226 Exercises 6.6 226 1 226 2, 3, 4 227 5, 6, 7 228 8 229 9, 10, 11 230 12, 13 231 Exercises 7.7 231 1, 2 231 3 232 4 233 5. 234 6 236 7, 8 237 9, 10 238 B. Table of Laplace Transforms 240 C. Linear Spaces 244 C,1 Linear Algebra 244 C.2 Gramm-Schmidt Orthonormalisation Process 256 Bibliography 258 Index 260 This Book Is A Self-contained Introduction To Laplace Transforms And Fourier Series; Emphasising The Applications Of Laplace Transforms Throughout, The Book Also Provides Coverage Of The Underlying Pure Mathematical Structures. Alongside The Laplace Transform, The Notion Of Fourier Series Is Developed From First Principles. Exercises Are Provided To Consolidate Understanding Of The Concepts And Techniques, And Only A Knowledge Of Elementary Calculus And Trigonometry Is Assumed. For Second And Third Year Students Looking For A Rigorous And Practical Introduction To The Subject, This Book Will Be An Invaluable Source. The Laplace Transform -- Further Properties -- Convolution And The Solutions -- Fourier Series -- Partial Differential Equations -- Fourier Transforms -- Complex Variables And Laplace Transforms -- Appendices: A: Answers To Exercises -- B: Table Of Laplace Transforms -- C: Linear Spaces. P.p.g. Dyke. Includes Bibliographical References (p. 243-244) And Index. "Emphasising the applications of Laplace transforms throughout, the book also provides coverage of the underlying pure mathematical structures. Alongside the Laplace transform, the notion of Fourier series is developed from first principles. Exercises are provided to consolidate understanding of the concepts and techniques, and only a knowledge of elementary calculus and trigonometry is assumed. For second and third year students looking for a rigorous and practical introduction to the subject, this book will be an invaluable source."--BOOK JACKET This introduction to Laplace transforms and Fourier series is aimed at second year students in applied mathematics. It is unusual in treating Laplace transforms at a relatively simple level with many examples. Mathematics students do not usually meet this material until later in their degree course but applied mathematicians and engineers need an early introduction. Suitable as a course text, it will also be of interest to physicists and engineers as supplementary material. As a discipline, mathematics encompasses a vast range of subjects.