Construction Notes

These notes are reminders of issues that have come up in building this website. They may appear and disappear as the site develops. If you think you can help, feel free to email me at nithardt@uw.edu.

Missing References

W&W have included a lot of references, all of which are old enough to be in the public domain. For many, I have found and linked to on-line versions. For some references, I have had no luck finding. This is a running list of such references.

Chapter 1.

All found

Chapter 2.

  1. this result (divergence of harmonic series) was noticed by Leibniz in 1673.
  2. the expressions convergent and divergent were introduced by James Gregory, Professor of Mathematics at Edinburgh, in the same year (1668).
  3. he (Newton) investigated the convergence of hypergeometric series in 1704.
  4. It (the Bolzano-Weierstrass theorem) seems to have been known to Cauchy.
  5. These methods (of summing double series) are due to Cauchy.
  6. The convergence of the product in which a_{n−1}=−1/n2 was investigated by Wallis as early as 1655.
  7. (de la Vallee Poussin, Mém. de l’Acad. de Belgique, LIII. (1896), no. 6.) Example 10.

Chapter 3.

  1. One suggestion (for visualizing complex functions) (made by Lie and Weierstrass) is to use a doubly-manifold system of lines in the quadruply-manifold totality of lines in three-dimensional space.
  2. Cauchy shewed that if a real function f(x), of a real variable x, satisfies the precise definition (of continuity), then it also satisfies what we have called the popular definition; …. But the converse is not true, as was shewn by Darboux.

Chapter 4.

  1. The formula for differentiation of integrals containing a parameter was given by Leibniz, without specifying the restrictions laid on f(x,α). (in marginnote).

Chapter 7.

  1. One of the miscellaneous exercises lists “Gambioli, Bologna Memorie, 1892.” as a reference.

    I was able to find Memorie della R. Accademia delle scienze dell’Istituto di Bologna (1892) but there is no mention there of Gambioli.

Chapter 8.

  1. The definition of asymptotic expansion "is due to Poincaré. Special asymptotic expansions had, however, been discovered and used in the eighteenth century by Stirling, Maclaurin and Euler."

Appendix

  1. In footnote to “ … an equation first given by Newton.” it says the series definition of \(e^x\) was also given both by Newton and by Leibniz in letters to Oldenburg in 1676.

W&W have given references for some examples, but many of these references are almost useless, like (Cauchy.). There are some, like (Trinity, 1904.), (Peterhouse, 1906.), or (Math. Trip., 1906.), which appear to reference specific college exams. I have found no on-line source for these, however.

Unresolved Examples

Here, I am keeping a running list of examples that I have been unable to solve or, as for starred items, I have only an ugly or over complicated solution. I am open to an exchange of hints or ideas, but please do not e-mail unsolicited solutions.

Chapter 2.
15*
21