Number 1, November 1996.
Number 1, November 1996.
James Davenport, University of Bath (United Kingdom) J.H.Davenport@maths.bath.ac.uk
My first thought on seeing this book was ``I wish I had written it''
- it resembles my lecture notes on the subject, but is better and
more complete.
After an introductory chapter on algebra, the chapter on rational
function integration makes good use of recent developments, e.g. the
Lazard-Rioboo-Trager algorithm for computing the transcendental part
by resultants. Rioboo's algorithm for producing continuous
anti-derivatives of real rational functions is also discussed, and its
importance properly pointed out.
The differential algebra chapter is fairly standard, though more
attention than usual is given to arbitrary monomial (a transcendental
is a monomial over
if
) extensions rather than
just to logarithmic and exponential ones. There is also a useful
chapter on orders, localizations and residues, paving the way for an
extension of the theory to algebraic extensions (which is not covered
in this book, though a good set of references is given).
Chapter 5 is devoted to the theory of elementary and Liouvillian
functions. The author makes a useful extension to the standard
presentation by working with primitives (integrals) and
hyperexponentials (exponentials of integrals), rather than just
logarithms and exponentials, at the beginning, and wherever possible
thereafter, e.g. in explaining the Hermite reduction and in the
residue criterion (if any residues at irreducibles are not constant,
then the integral cannot be elementary). Figure 5.1 is an
excellent schematic of the integration algorithm, and should be used
by everyone lecturing on the subject. This chapter also shows the
importance of recognising the ``limited integration problem'',
i.e. integration with a fixed set of new admissible functions, as a
fundamental building block in the process. This chapter also
introduces hypertangents ( is a hypertangent over
if
) as fundamental objects, and presents an integration
algorithm for
if
is a hypertangent, subject to being able
to solve a coupled differential system (instead of the single Risch
differential equation that an exponential gives rise to) over
. This can be used to integrate trigonometric functions directly,
without use of complex exponentials.
The next chapter looks at the Risch differential equation problem that
arises when integrating exponentials: solve for
. Again, as much as possible is done for arbitrary monomials, and
hypertangents are treated as first-class objects. The presentation is
based on Rothstein's work, and is the first widely-accessible account
of this method.
There is then a chapter on various parametric problems, i.e. problems
with unknown constants to be determined, notably the parametric Risch
differential equation and the limited integration problem. Much of the
detail of this chapter is new, previous authors having said ``
can easily be generalized'' or words to that effect. Chapter 8, on the
coupled differential system, is also largely new. The author points
out that what arises is not an arbitrary 2x2 system, but really the
real and imaginary parts of a Risch differential equation, as one
would expect from the complex exponential formulation of tangents.
The book concludes with a chapter on structure theorems, presenting
both the standard Risch structure theorem and the Rothstein-Caviness
structure theorem.
On the negative side, the author says of the Horowitz-Ostrogradsky
method ``it does not generalize as easily as the Hermite reduction to
larger classes of functions''. This is certainly true, but the author could
have given some references to the ``Risch-Norman'' literature to
explain the problems, and to show what is known about integrating
higher functions by extensions of the Horowitz-Ostrogradsky
method. Apart from this, the bibliography is pretty complete, and I
learnt of some useful new references.
The algorithms are generally well-presented in pseudo-code, and the
index serves also as an index of algorithms. I would have liked to see
an index of notation as well, though. Most of the chapters have
exercises at the end, though towards the end these tend to tail off,
probably a consequence of the increasing difficulty and newness of the
material.
In sum, the book does what it sets out to do, does it well, and should
be on the bookshelf of every implementer or teacher. For the latter,
the author indicates various ways through the material, depending on
the aims and background of the course being taught.
Contents