Classics in Differential Equations
Advanced books are difficult to read. It takes a good deal of time to reading
classics. Classics attract many
talented scientists to their past glory. After their authors die, it becomes difficult for others to update the original work.
Considering the time required to study these classics, we must know their merits
and shortcomings so that we may focus on their strength and spend less time on
their outdated material and other drawbacks. In addition, we desire to find new
blood to compensate for their shortcomings. For example, even today there are
scientists reading [Wat1]. The book was the best in the past in the sense of
being well updated. Even though it is no longer current, it is still considered
a classic simply because few mathematicians today have sufficiently
broad knowledge to wrote a better book. We may facilitate our study on the
subjects that were difficult in the past by viewing them from another perspective (e.g.,
Riemann's theorems [Wat1, p.182, l.-5; Zyg, vol. 1,
p.352, Theorem 7.4] can be considered special cases of the statement given in [Ru2,
p.90, l.5]). We may also
unify subjects discussed separately in the past (e.g., Infinite integrals
[Wat1, pp.69-74, §4.4] and improper integrals
[Wat1, pp75-77, §4.5] may be considered special cases of the Lebesque's
integral [Ru2, p.20, l.-5]). Thus, an improper use of
classics can lead to a tremendous waste considering the enormous time and the
large number of people involved.
- Chap. IV: Replace infinite and improper integrals [pp.69-77,
§4.431, §4.44 & §4.5] with the Lebesque
integrals [Ru2, chap. 1].
- Chap. VI: Replace [pp.112-119, §6.2,
§6.23 and §6.24] with
[Gon, pp.680-730, §9.10 &
Chap. VIII: Replace [§8.3,
§8.31 and §8.32]
with [Guo, p.32, l.3-p.34, l.4]; Replace §8.43 with [Zyg, vol. 1, chap. III].
- Chap. XI: The formula given in [Wat1, p.211, l.13] should be corrected as
f(x) = ò[0, +¥] cos (xt)
f(t)dt. See [Tit, p.2, (1.1.4)].
- Chap. XII: In [p.257, l.1-l.5], we should use the concept of the
Riemann surface to discuss the argument changes of t and t-1
(rather than 1-t) along the path
of integration given in the figure on p.257.
- Chap. XIV:
1. The 24 solutions given in [Wat1, §14.3] lack a useful strategy to
classify them. [Wat1, §14.3] should be replaced with [Guo, §4.3].
2. Although [Wat1, p.287, l.-2-l.-1] is a good observation, [Wat1, §14.5] should be replaced by [Guo, §4.6]. However, [Guo, p.155, (6)] should be replaced by [Wat1, p.279, l.4].
- Chap. XV: Replace [Wat1, §15.6] with [Hob, chap. V. §116-§125].
- Chap. XXI: Replace [§21.71,
§21.711 and §21.712] with [Sak, chap. VIII,
- [p.123, l.17-p.124, l.10] should be replaced by [Cod, p.86, Theorem 6.3].
Asymptotic development of solutions