In this book I have tried to present a rigorous account of the elementary
theory of the Laplace transform, besides giving a description of some of
its more useful applications. The book is primarily intended for mathe-
matics students, though it is hoped that it will also be of value to students
of physics or engineering who wish to understand more of the mathe-
matical tools they use. I have taken as my standard the level of a course
comparable with those required in Methods of Mathematical Physics
for the Special Mathematics Degree, or the Postgraduate Diploma in
Mathematics of the University of London. I have been associated with
both courses at my present college. I have therefore assumed a knowledge
of mathematics which a student taking such a course could ordinarily
be expected to possess. In particular I have assumed that the reader is
familiar with Cauchy's theorem and its immediate consequences and of
the simple properties of the factorial function z! (used in preference to
the Gamma function r(z)=(z-l)!), the Legendre function Pv(z) and
the Bessel function of the first kind lv(z). Other functions are referred
to, but then I hope the passages are self-contained, and need no previous
knowledge.
Author(s): M. G. Smith
Publisher: Van Nostrand Reinhold
Year: 1966
Language: English
Pages: 133
Preface
CHAPTER 1 THE D-OPERATOR 1
1.1 Operational Techniques 1
1.2 Simple D-algebra 2
1.3 The Inverse Operator 4
1.4 Expansion of the Inverse Operator 6
Worked Examples 7
Exercises 10
Answers 11
CHAPTER 2 THE LAPLACE INTEGRAL 12
2.1 Conditions for Existence 12
2.2 Abscissae of Convergence 13
2.3 Elementary Transforms 14
2.4 Simple Applications 17
Worked Examples 18
Exercises 20
Answers 21
CHAPTER 3 SIMPLE PROPERTIES OF THE LAPLACE TRANSFORM 22
3.1 Linear Property 22
3.2 Change of Scale 22
3.3 Shift of the Origin 23
3.4 Periodic Functions 23
3.5 Multiplication by a Simple Power 24
3.6 Expressions involving Inverse Powers of t 25
3.7 Transform of Integrals 25
3.8 Transform of an Integral Function 26
3.9 Examples illustrating 3.8 27
3.10 Behaviour for Large Isl 28
Exercises 28
Answers 29
CHAPTER 4 THE INVERSE INTEGRAL 30
4.1 The Inversion Problem 30
4.2 The Fourier Transform 30
4.3 The Inverse as a Contour Integral 32
4.4 Examples of Inversion 35
Exercises 40
CHAPTER 5 THE CONVOLUTION OR FALTUNG INTEGRAL 43
5.1 The Repeated Integral 43
5.2 Simple Application to Differential Equations 45
5.3 A Generalization of 5.2 46
5.4 Transform of a Product 48
5.5 A Simple Illustration 49
Exercises 49
Answers 50
CHAPTER 6 APPLICATION TO ORDINARY LINEAR DIFFERENTIAL EQUATIONS 51
6.1 Limitations of the Technique 51
6.2 The Bessel Equation 52
6.3 Another Related Equation 53
6.4 The Laguerre Equation 53
6.5 The Hermite Equation 57
6.6 The Laplace Linear Equation 58
6.7 A Further Illustrative Example 60
Exercises 61
Answers 62
CHAPTER 7 PARTIAL DIFFERENTIAL EQUATIONS 64
7.1 Limitations of the Technique 64
7.2 A Simple Equation of the First Order 65
7.3 The Diffusion Equation 66
7.4 The Wave Equation 68
7.4.1 The infinite string 68
7.4.2 The finite string 69
7.4.3 The axially-symmetric equation 70
7.4..4 The equation in three dimensions 71
Exercises 73
Answers 74
CHAPTER 8 LINEAR INTEGRAL EQUATIONS
8.1 Types of Integral Equations
8.2 The Difference Kernel
8.2.1 The equation of the first kind
8.2.2 The equation of the second kind
8.3 The Wiener-Hopf Technique
8.4 The Milne Equation
Exercises
Answers
CHAPTER 9 LINEAR DIFFERENCE EQUATIONS
9.1 The Difference Equations
9.2 Transform of a Step Function
9.3 Simple Applications Worked Examples
9.4 Inhomogeneous Equations Worked Examples
9 .5 Variable Coefficients
Worked Examples
Exercises
Answers
CHAPTER 10 ASYMPTOTIC FORMULAE
10.1 Elementary Asymptotic Results
10.2 Asymptotic Form of a Transform for Large (s(
10.3 Asymptotic Series of a Transform for Large Isf
10.4 Application to the Exponential Integral
10.5 Asymptotic Behaviour off(t)
10.5.1 Single-valued function
10.5.2 Function with a branch point Worked Examples
10.6 Terms involving Logarithms
A Simple Example
Exercises
Answers
APPENDIX 1 THE FOURIER INTEGRAL THEOREM 107
APPENDIX 2 THE EXPONENTIAL INTEGRAL 112
APPENDIX 3 OPERATIONAL FORMULAE 115
APPENDIX 4 TABLE OF THE MORE COMMON LAPLACE TRANSFORMS 118
References 120
Index 121
