## An Introduction to Laplace Transforms and Fourier SeriesThis 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 |

245 | |

247 | |

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### Common terms and phrases

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