On the critical values of the sums of periodic series. Offprint from Transactions of the Cambridge Philosophical Society, Vol. VIII, Part V, 1848. THE "GIBBS PHENOMENON". PRESENTATION COPY INSCRIBED BY STOKES

STOKES, George Gabriel

Publication Date: 1848

About this Item

First edition, very rare inscribed offprint (COPAC lists three copies) of the discovery of the "Gibbs phenomenon" and the invention of the notion of uniform convergence of infinite series. "An understanding that the convergence of a series of functions needs to be carefully handled developed only in the late 1840s, simultaneously but independently in the work of Phillip Seidel, a former pupil of Dirichlet, and George Stokes in Cambridge. Both realized that for an infinite sum of continuous functions to be continuous, convergence must be of a particular kind, which Seidel (in 1848) described as 'not arbitrarily slow' and Stokes (in late 1847) as 'not infinitely slow'. Such ideas foreshadowed the later idea of uniform convergence, where a series of functions not only converges at every point, but does so at a uniform rate over a range of values. Stokes paper of 1847 was entitled 'On the critical values of the sums of periodic series'. His discussion of continuity in §1 shows that even by 1847 the Eulerian definition of a continuous function as one that can be expressed by a single formula had not entirely fallen out of use, but Stokes himself adopted the Bolzano-Cauchy definition. In the same opening section, Stokes separated convergent series into two classes, either essentially or accidentally convergent; in modern terms absolutely or conditionally convergent. His definition of infinitely slow convergence came only much later in the paper, in §38. . . . The modern definition of uniform convergence was formulated a few years after the appearance of Stokes' paper, by Weierstrass in Berlin" (Stedall, Mathematics Emerging, A Source Book 1540-1900, pp. 527-528). If a Fourier series converges uniformly, its sum must be continuous. So the Fourier series of a discontinuous function cannot converge uniformly. This is reflected in the so-called "Gibbs phenomenon", rediscovered by Gibbs half a century later. It refers to the behaviour of the Fourier series of a function near a jump discontinuity - the Fourier series overshoots on the high side of the jump and undershoots on the low side, the amount of the overshoot and undershoot being about 9% of the gap. As Michael Berry points out ['Stokes' phenomenon; smoothing a victorian discontinuity', Publications mathématiques de l I.H.É.S. 68 (1988), p. 214], this phenomenon was described by Stokes in the present paper. It was also discovered independently by Henry Wilbraham in the same year, and is sometimes known as the Wilbraham-Gibbs phenomenon, though Stokes is rarely credited. "Th[is] memoir is remarkable in many respects, containing a general discussion of the possible modes of convergence, both of series and of integrals, far in advance of the current ideas of the time" (From Kant to Hilbert, Ewald, ed., p. 1235). Before going to Cambridge, W. F. L. Fischer studied in both Berlin and Paris where he took classes by Dirichlet, Liouville and Poisson. At Cambridge, Fischer, known as Franz to his friends, studied with William Thomson, later Lord Kelvin, under the famous coach William Hopkins. Fischer graduated fourth Wrangler in 1845, becoming a Fellow of Clare College in 1847. On the departure of John Couch Adams in 1859, Fischer became Professor of Mathematics at St. Andrews, a position he held until 1877. 4to, pp. [1-2], 3-53. Original plain wrappers. Inscribed on title 'Professor Fischer from the author'.

Seller Inventory # ABE-1470927063077

Bibliographic Details

• Title: On the critical values of the sums of periodic series. Offprint from Transactions of the Cambridge Philosophical Society, Vol. VIII, Part V, 1848. THE "GIBBS PHENOMENON". PRESENTATION COPY INSCRIBED BY STOKES
• Publication Date: 1848
• Binding: Soft cover
• Edition: 1st Edition
• Condition: Very Good
• Signed: Inscribed by Author(s)