id: 06088712
dt: b
an: 2013e.00623
au: Zemyan, Stephen M.
ti: The classical theory of integral equations. A concise treatment.
so: Basel: Birkhäuser (ISBN 978-0-8176-8348-1/hbk; 978-0-8176-8349-8/ebook).
xiii, 344~p. (2012).
py: 2012
pu: Basel: Birkhäuser
la: EN
cc: I95
ut: integral equations; Fredholm equations; Volterra equations; textbook;
Hermitian kernels; eigenvalue; eigenfunction; nonlinear equations;
singular kernels; Laplace transform; Fourier transform
ci: Zbl 0078.09404
li: doi:10.1007/978-0-8176-8349-8
ab: This book covers the classical theory of linear, scalar Fredholm and
Volterra equations. A brief resumé follows. Chapter 1 deals with
Fredholm equations with separable kernels and gives the classical
Fredholm theorems. Chapter 2 considers the Fredholm theorems for
general nonseparable complex-valued kernels $K(x,t)$. Chapter 3
presents the theory for Fredholm equations with Hermitian kernels. The
classical eigenvalue and eigenfunction theory is given, including the
$L_2$-theory. These first three chapters comprise half the book.
Chapter 4 gives some elementary Volterra theory. Chapter 5 presents
techniques for converting differential equations to integral equations
and vice versa. Chapter 6 looks at nonlinear equations and includes
some illustrative examples. Chapter 7 lists some equations with
singular kernels and mentions Laplace and Fourier transforms. A brief
look at linear systems of integral equations makes up the final Chapter
8. To conclude the book, the author offers brief French, German and
Italian mathematical vocabularies and sample translations. All chapters
include numerous simple examples, some explicitly solvable, some to
which numerical methods are applied. No applications to, e.g., physics
or engineering are given. In its content, the book is reminiscent of
the classical work by {\it F. G. Tricomi} [Integral equations. Pure and
Applied Mathematics, Vol. 5. New York: Interscience Publishers (1957;
Zbl 0078.09404)]. All integrals in this volume are assumed to be
Riemann integrals, not Lebesgue integrals. Perhaps unavoidably, this
entails some errors as the statement that the set of functions having
finite square Riemann integrals $\int|f(t)|^2\,dt$ is complete in the
$L_2$-norm. The book is somewhat too elementary for the intended
audience of advanced undergraduate or early graduate students of
mathematics but may be useful for engineering students wishing to learn
the fundamentals of integral equation theory.
rv: Stig-Olof Londen (Aalto)