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fluid mechanics, can be found in Kaplun (1967); this is advisable reading only for those
with a deep interest in fluids. The construction of asymptotic solutions to ordinary
differential equations (that is, in the absence of a small parameter) is described in Wasow
(1965) and also in Dingle (1973); this latter book provides a very extensive analysis
of asymptotic expansions, their properties and how to construct useful forms of them
that provide the basis for numerical estimates.
There are various texts that describe the techniques for finding asymptotic ex-
pansions of functions defined, for example, by differential equations or as integrals;
excellent examples in this category are Erdelyi (1956), Copson (1967) and Murray
(1974). But the outstanding text has got to be Olver (1974), for its depth and breadth;
furthermore, this provides an excellent reference for the behaviour of many standard
functions (and the results are presented, comprehensively, for functions in the complex
plane). Finally, we mention two texts that discuss the properties of divergent series:
Hardy (1949) and Ford (1960). The former has become a classical text; it covers a lot
of ground but is written in a pleasant and accessible style—it cannot be recommended
highly enough.

EXERCISES
Q1.1 Maclaurin expansions I. The following real functions are defined for suitable real
values of x. Use your knowledge of Maclaurin expansions to find power-series
representations of these functions as giving the first three terms in
each case; state those values of x for which the expansions are convergent.
(Some of these may not be expressible wholly in terms of integral powers of x.)

(a) (b) (c) (d)
(e) (f) (g)
Q1.2 Maclaurin expansions II. Repeat as for Q1.1, but for each of these series also find
the general term in the expansion.

(a) (b) sin(l + x); (c)
Q1.3 Exact solutions of ODEs. Find the exact solutions of these ordinary differential
equations and explore their properties as (for example, by sketching
or plotting the solutions for various You may wish to investigate how an
appropriate approximation, for might be obtained directly from the
differential equation (The over-dot denotes the derivative with respect to t,
time, for problems in the prime represents the derivative with respect
to x )
(a) with
(b) with
(c) with

(d) with