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Term-by-Term Integration of Infinite Series

Published online by Cambridge University Press:  03 November 2016

H. S. Carslaw
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1. The English student of Mathematics finds in his text-books a proof that the uniform convergence of a series of continuous functions is a sufficient condition for term-by-term integration. He knows that there are convergent series which cannot be integrated term-by-term,f and that there are series, not uniformly convergent in the interval of integration, which can be thus integrated. The last remark applies particularly to the Fourier’s Series of an ordinary function with discontinuities.

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Research Article
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The Mathematical Gazette , Volume 13 , Issue 191 , December 1927, pp. 437 - 441
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Copyright © Mathematical Association 1927

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References

page note 437 * As a matter of fact if the functions are integrable and the series converges uniformly, the sum is integrable and term-by-term integration is allowable. But this is much harder to prove than the standard theorem about a series of continuous functions. Cf. Hobson, Theory of functions of a Real Variable, i.(2nd edit. 1921) 446.

page note 437 † When Sn =2n 2xe-n2x2 , lim sn = 0 and lim sn dx = 0 = lim s n dx.

Throughout this paper sn or Sn(x) stands for the sum of the first n term of a series.

page note 437 ‡ When

Again

page note 437 § See below, § 2, concluding sentence.

A simple proof is given in de la Valleé Poussin’s Cours d’Analyse Infinitésimale, 2 (4th ed. 1922), 106, that the Fourier’s Series of a function which is absolutely integrable can always be integrated term-by-term, even when the series itself does not converge.

page note 437 ∥ Hobson, loc. cit. 2 (2nd ed. 1926), 303.

page note 437 ¶ Cf. e.g. Proc. London Math. Soc. (2), 9 (1910), p. 316 and p. 464.

page note 438 * Cf. e.g. Bromwich, Theory of Infinite Series (2nd ed. 1926), footnote on p. 134, and Ex. 22 on p. 144. I wonder if an elementary proof of the first part of this example is possible. For the second part, when the functions are integrable, see below, §§ 3. 4.

page note 438 † Rend. Acc. Line. (4), 1 (1885) 537, and, more fully, in Mem. 1st. Bologna (5), 8 (1900), 723. Osgood, independently of Arzela, gave the case for continuous functions in Amer. Journal of Math. 19 (1897), 155.

page note 438 ‡ Bieberbach, “über einen Osgoodschen Satz aus der Integralreehnung,” Math. Zeitsehrift, 2 (1918), 155.

page note 438 § Landau, “Ein Satz über Riemannsche Integrale,” ibid. p. 350.

page note 439 * Cf. e.g. Hobaon, loc. cit. 2 (2nd ed. 1916), p. 512.

This result can also be deduced from the Gibbs’ Phenomenon in Fourier’s Series. Cf. my Fourier’s Serien and Integrals, eh. ix. (2nd ed. 1921).

page note 439 † See also footnote §, p. 1.

page note 439 ‡ The proof of the theorem on sets of intervals which is used here win be found in the next section.

page note 440 This theorem was first proved by Arzela, loc. cit. The proof given here is due to Bieberbach, loc. cit. The theorem is a special case of a more general theorem in sets of points due to W. H. Young, to be found in Hobson’s Theory of Functions, 2 (2nd ed. 1926), § 136, see especially p. 177.