1 1 1 1 72 cos x cos 2 x cos 3.2 12 12 + 22 - 32 t...; (113) and if in (112) c is replaced by its value, 12 + 22 + 32 + ... (114) Also taking the members of (104) to be the element-functions of a definite integral for the limits x and 0, where x is not greater than 7, za ToX 4 sin x sin 3x sin 53 ) Ex. 2. Let f(x) = l; then the theorems contained in (92) and (93) are evidently identities. But taking (97), sin x sin 3x sin 52 (117) for all values of x between 0 and 7 exclusively; and =-ä, for all values of x between - 1 and 0 exclusively; and = 0, when x = -1, = 0, and = 1. Ex. 3. Let f(x) = sin x. Then by the general theorem in (89) we have also differentiating (126), sin mx sin x 2 sin 2 x 3sin 3 x 2 sin ma = 12 — m2 ~ 22 - m2 + 32 – m2 ...; (120) which is true for all values of x from — 5 to a exclusively. Ex. 5. Let f(x) = emx + e-mx. Then from (96), ens + e-mz 1 2m ( cos x cos 2 x cos 3x et - e-ma m 1m2 +1 m3 + 22 m2 + 32 "'S which is true for all values of x from – a to a inclusively. If x = 1, (129) gives emite-mi 1 1 1 1 2memo-e-mi = 2m2 + m2 I 12+ m2 1 92 + m2 1 22+ .... (130) Hence, if m=0, 1 + 23 + 32 + ... = (131) Also differentiating (129), Teme-e-mot sin x 2 sin 2 x 3 sin 3x 2 em-e-ma = m2 +1- m2 + 22 + m + 32 -...; (102) which is true for all values of x from – to a exclusively. And if in (132) x is replaced by – X, then 70+ (-2) - e-m(7—) sin x 2 sin 2. 3 sin 3.2 2 eme ce-mim +1 + m2 + 22 + m +32 7. 22 + ...; (133) which is true for all values of x from 0 to 27 exclusively. 203.] The preceding examples are sufficient to illustrate the general theorems of periodic series. Many others will be found in a memoir entitled “ Die Lagrangesche Formel und die Reihensummirung durch dieselbe,” by J. Dienger, Crelle's Journal, Vol. XXXIV, p. 75. The theory however admits of consideration from another point of view, and thereby has another important application; viz. that to discontinuous functions. The fact has been explained in Art. 199, and it is now only requisite to exhibit some cases of its application. Herein I shall take the theorems given in Art. 201, of which the period is 2c instead of 21 as in the preceding examples. Ex. 1. Find a periodic function of x, whose period is 2c, which is equal to 1 for all values of x between 0 and c, and is equal to -1 for all values of x between –c and 0. As the values of f (x) in this case are to be equal, but of oppoPRICE, VOL. II. Nn site signs for the positive and negative values of x, we must take the theorem given in (101); and we have . NTX c. naz *** dz, . sin 41. TX1 . 3 1 X 1.5 TX ) {sin + zsina* + sina +...}. (134) Also the value of this series, at the limits 0 and c, = 0. Thus we have the following geometrical interpretation. Let the righthand member of (134) express the ordinate of a locus; then that locus between x = -c and x = 0, both exclusively, is a straight line at a distance = -1 from the x-axis; when x = 0, it is a point on the x-axis ; between x = 0 and x = c, both exclusively, it is a straight line at a distance = 1 from the x-axis ; and when x = c, it is a point on the x-axis. And as we have traced the locus through a complete period, these lines and points are repeated ad infinitum in both directions. Ex. 2. Find a periodic function of x, whose period is 2c, which is equal to – mx, for all values of x between –c and 0; and = mx for all values of x between 0 and c. As the values of f(x) in this case are to be equal, and of the same sign for equal, positive, and negative values of x, we must apply the series of cosines given in (100); whence we have cm 4cm s. TTX 1 378 1 5 TX f(x) = 2- 2 {cos +zz cosĆ +52cos +... };(135) and this is true at the limits as well as for other values of x. This admits of a remarkable geometrical interpretation; let the right-hand member of (135) express the ordinate of a locus; then that locus consists (1) of the portion of the straight line, whose equation is y = – mx, contained between x = -c and x = 0, the line terminating abruptly at these values of x; (2) of the portion of the line whose equation is y = mx, contained between x = 0, and x = c, this line also terminating abruptly at these values of x. Also these lines are repeated ad infinitum in both directions. So that the locus consists of a series of straight lines forming the sides of isosceles triangles, the lengths of whose bases are 2 c placed continuously along the x-axis on its positive side, and whose base angle = tan-1 m. Ex. 3. Find a periodic function of x whose period is 2c, which is equal to mx for all values of x between 2c and 0. which is true for all values of x between 0 and 2c exclusively; and = 0, when x = 0; and = mc, when x = 2c. Thus, if (136) is taken to express the ordinate of a locus, that locus consists of (1) the length of a straight line, whose equation is y = mx, contained within the limits x = 0 and x = 2c; (2) of a point x = 2c, y = mc, which is a discontinuous point ; (3) of similar lines and points repeated ad infinitum in both directions. Ex. 4. Find a periodic function of X, whose period is 2c, which = a for all values of x between x = 0 and x = k; and = b for all values of x between x = k and x = 2c. and at the limits, f(0) = f(c)= f (2c) = 0. This is the result already found in example 1. The geometrical interpretation is similar to that of that example. Ex. 5. Determine S (x), a periodic function of x, whose period is 27, which = x from x = 0 to x = a; = a from x = a to 1 = ti-a; and = 1- x from x = a - a to x = ; and is also such that f(-x) = -f(x). |