文档介绍：To my parents
v
It is customary to begin courses in mathematical engineering by ex-
plaining that the lecturer would never trust his life to an aeroplane
whose behaviour depended on properties of the Lebesgue integral.
It might, perhaps, be just as foolhardy to fly in an aeroplane de-
signed by an engineer who believed that cookbook application of
the Laplace transform revealed all that was to be known about its
stability.
. Korner¨
Fourier Analysis
Cambridge University Press
1988
vii
Preface
The Laplace transform is a wonderful tool for solving ordinary and
partial differential equations and has enjoyed much ess in this
realm. With its ess, however, a certain casualness has been bred
concerning its application, without much regard for hypotheses and
when they are valid. Even proofs of theorems often lack rigor, and
dubious mathematical practices are not mon in the literature
for students.
In the present text, I have tried to bring to the subject a certain
amount of mathematical correctness and make it accessible to un-
dergraduates. To this end, this text addresses a number of issues that
are rarely considered. For instance, when we apply the Laplace trans-
form method to a linear ordinary differential equation with constant
coefficients,
(n) (n−1)
any + an−1y +···+a0y
f (t),
why is it justified to take the Laplace transform of both sides of
the equation (Theorem )? Or, in many proofs it is required to
take the limit inside an integral. This is always frought with danger,
especially with an improper integral, and not always justified. I have
plete details (sometimes in the Appendix) whenever this
procedure is required.
ix
x Preface
Furthermore, it is sometimes desirable to take the Laplace trans-
form of an infinite series term by term. Again it is shown that
this cannot always be done, and specific sufficient conditions are
established to justify this operation.
Another delicate problem in the literature has b