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we must know their merits and shortcomings so that we may focus on their strength and spend less time on their outdated material and other drawbacks.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 still many 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 Even though it is no longer current, it is still considered a classic simply because few mathematicians today have sufficiently broad knowledge to wrote a better book.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.
  1. [Wat1]
    1. Chap. IV: Replace infinite and improper integrals [pp.69-77, 4.4, 4.41, 4.42, 4.43, 4.431, 4.44 & 4.5] with the Lebesque integrals [Ru2, chap. 1].
    2. Chap. VI: Replace [pp.112-119, 6.2, 6.21, 6.22, 6.221, 6.222, 6.23 and 6.24] with [Gon, pp.680-730, 9.10 & 9.11].
      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].
    3. Chap. XI: The formula given in [Wat1, p.211, l.13] should be corrected as f(x) = [0, +] cos (xt) f(t)dt. See [Tit, p.2, (1.1.4)].
    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.
    5. 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].
    6. Chap. XV: Replace [Wat1, 15.6] with [Hob, chap. V. 116-125].
    7. Chap. XXI: Replace [21.71, 21.711 and 21.712] with [Sak, chap. VIII, 11].

  2. [Inc1]
    1. [p.123, l.17-p.124, l.10] should be replaced by [Cod, p.86, Theorem 6.3].
    2. 7.31: Asymptotic development of solutions