## Introduction to the Laplace TransformThe purpose of this book is to give an introduction to the Laplace transform on the undergraduate level. The material is drawn from notes for a course taught by the author at the Milwaukee School of Engineering. Based on classroom experience, an attempt has been made to (1) keep the proofs short, (2) introduce applications as soon as possible, (3) concentrate on problems that are difficult to handle by the older classical methods, and (4) emphasize periodic phenomena. To make it possible to offer the course early in the curriculum (after differential equations), no knowledge of complex variable theory is assumed. However, since a thorough study of Laplace. transforms requires at least the rudiments of this theory, Chapter 3 includes a brief sketch of complex variables, with many of the details presented in Appendix A. This plan permits an introduction of the complex inversion formula, followed by additional applications. The author has found that a course taught three hours a week for a quarter can be based on the material in Chapters 1, 2, and 5 and the first three sections of Chapter 7. If additional time is available (e.g., four quarter-hours or three semester-hours), the whole book can be covered easily. The author is indebted to the students at the Milwaukee School of Engineering for their many helpful comments and criticisms. |

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### Contents

Basic Properties and Applications | 1 |

12 The Variable s | 3 |

13 Laplace Transforms of Some Special Functions | 4 |

14 Some Basic Properties of the Laplace Transform | 10 |

15 Inverse Laplace Transforms | 18 |

16 Partial Fractions | 21 |

17 Differential Equations | 30 |

18 Applications | 33 |

32 The Residue Theorem | 100 |

The Complex Inversion Formula | 105 |

42 The Inversion Integral | 107 |

Convolutions | 111 |

52 Two Special Limits | 118 |

53 Applications | 119 |

Transforms with Infinitely Many Singularities | 129 |

Applications to Partial Differential Equations | 145 |

19 Differentiation and Integration of Transforms | 38 |

Further Properties and Applications | 43 |

22 The Second Translation Theorem | 45 |

23 Transforms by Graphical Addition | 50 |

24 The Unit Impulse Function | 55 |

25 Applications | 63 |

26 Transforms of Periodic Functions | 74 |

27 Applications | 83 |

Sketch of Complex Variable Theory | 93 |

72 The Diffusion Equation | 152 |

73 The Vibrating String | 159 |

74 The Complex Inversion Formula Again | 164 |

More on Complex Variable Theory | 169 |

Table of Laplace Transforms | 179 |

185 | |

Answers to Exercises | 187 |

201 | |

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