# An Introduction to Laplace Transforms and Fourier Series

Springer Science & Business Media, Oct 27, 2000 - Mathematics - 250 pages
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.

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### Contents

 The Laplace Transform 1 12 The Laplace Transform 2 13 Elementary Properties 5 14 Exercises 11 Further Properties of the Laplace Transform 13 22 Derivative Property of the Laplace Transform 14 23 Heavisides Unit Step Function 18 24 Inverse Laplace Transform 19
 47 Exercises 108 Partial Differential Equations 111 52 Classification of Partial Differential Equations 113 53 Separation of Variables 115 54 Using Laplace Transforms to Solve PDEs 118 55 Boundary Conditions and Asymptotics 123 56 Exercises 126 Fourier Transforms 129

 25 Limiting Theorems 23 26 The Impulse Function 25 27 Periodic Functions 32 28 Exercises 34 Convolution and the Solution of Ordinary Differential Equations 37 33 Ordinary Differential Equations 49 331 Second Order Differential Equations 54 332 Simultaneous Differential Equations 63 34 Using Step and Impulse Functions 68 35 Integral Equations 73 36 Exercises 75 Fourier Series 79 42 Definition of a Fourier Series 81 43 Odd and Even Functions 91 44 Complex Fourier Series 94 45 Half Range Series 96 46 Properties of Fourier Series 101
 63 Basic Properties of the Fourier Transform 134 64 Fourier Transforms and Partial Differential Equations 142 65 Signal Processing 146 66 Exercises 153 Complex Variables and Laplace Transforms 157 73 Complex Integration 160 74 Branch Points 167 75 The Inverse Laplace Transform 172 76 Using the Inversion Formula in Asymptotics 177 77 Exercises 181 Solutions to Exercises 185 Table of Laplace Transforms 227 Linear Spaces 231 C2 GrammSchmidt Orthonormalisation Process 243 Bibliography 245 Index 247 Copyright