Abstract
Dirichlet’s papers proved rigorously and beyond dispute what Abel had already shown by an example: Fourier series can represent discontinuous functions. This established that there is an entire class of convergent series that are composed of continuous functions, but for which the sum was not continuous and that consequently contradicted Cauchy’s theorem on the continuity of the sum of a series of continuous functions.
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Notes
- 1.
Chapter closely follows Bottazzini § 5.4. The Uniform Convergence of Series, but draws on later historical literature, notably Bråting (2007) and Viertel (2014). A brief summary of the theorems relating to the properties of convergent sequences of integrable, continuous, or differentiable functions is provided in Appendix B.
- 2.
See Seidel, (1847, 35), quoted in Bottazzini, The Higher Calculus, 202.
- 3.
See Seidel (1847, 36–37), quoted in Bottazzini, The Higher Calculus, 202–203.
- 4.
See Seidel (1847, 37), quoted in Bottazzini, The Higher Calculus, 203.
- 5.
Grattan-Guinness argued on the basis of similarities between the two papers including the citations and the fact that Björling had also published some results on convergence teats in Liouville’s Journal that Cauchy would surely have seen. It is no argument against this view that Cauchy did not mention Björling by name in his (1853).
- 6.
See Björling (1846, 21) quoted in Bråting (2007, 520).
- 7.
See Gudermann (1838, 251–252). However, as Viertel has noted (2012, 142–144), Gudermann seems to have used this and other terms about the nature of convergence impressionistically and without defining them; other terms refer to the rate of convergence, which may well have meant the number of terms n that must be considered before a sequence is within a given \(\varepsilon \) of its limit.
- 8.
See Weierstrass (1841b, 68–69).
- 9.
See Weierstrass, ‘Zur Functionenlehre’, (1880). Today the result is obtained by applying the Heine–Borel theorem.
- 10.
He was also a descendant of the tradition begun by Peacock and Babbage, as can be seen by his use of the first difference operator, \(\Delta \).
- 11.
See Stokes (1849, 279), quoted in Bottazzini, The Higher Calculus, 205.
- 12.
See Stokes (1849, 281), quoted in Bottazzini, The Higher Calculus, 206.
- 13.
See Hardy (1918, 155), quoted in Bottazzini, The Higher Calculus, 206.
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Gray, J. (2015). Uniform Convergence. In: The Real and the Complex: A History of Analysis in the 19th Century. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23715-2_22
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