An Introduction to Laplace Transforms and Fourier Series

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