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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 Excerpt:...The coefficients at and 6„ or as we will call them, following a suggestion of Hurwitz, the Fourier's constants of might be determined by making use of the expression (5) for/)(x). They may be determined much more readily as follows by the method of undetermined coefficients. By differentiating the series (7) term by term first once and then twice, we get new trigonometric series which we will call (7') and (7"). Although we shall not attempt to make any statement as to whether these series represent the functions l'(x) and 4'"(x) or even as to whether they converge, we do know that their coefficients are the Fourier's constants of f'(x) and f"(x) respectively.t If, then, we multiply the terms of (7) by /9, those of (7') by 2a and add the two series thus obtained to (7"), we shall get a series whose coefficients are the Fourier's constants of £" + 2af' + $f, that is of f(x). If wc denote the Fourier's constants of f(x) by 7,, S;, so that These are linear equations for the determination of «, and ht in terms of the known quantities 7,, S(. The determinant of these equations is Cf. for instance the writer's Introduction to the Theory of Fuurier'i Series, these Annals, vol. 7 (1906), p. 109, Theorem II j nlso published separately by Harvard University. t This follows from the fact that f and p' are continuous, while )" la Unite for all values of x and has only a finite number of discontinuities in any Unite interval. Cf. the paper last referred to, p. 11(1, Theorem I. I Cf. p. 87, Theorems V, VI, of the paper last cited. This quantity does not vanish, since o?t 0, /9 0. Thus we get for at and 6 the values Substituting these values in the series (7), we get the Fourier's development of )x). This development may, by a...