FOURIER SERIES

infinite series involving cosines and sines of the succe ive multiples of a variable, i.e. series of the type. Ap + a, cos9 + a, cos29 + a3.c0s39-+... +b, sin?+6, sin26+b, sin3#+.,. A function of ¢ represented by a convergent series of this type is periodic, the period being 27; for every term has the same value for @+ 2k7 as for @, where k is any integer. The corresponding form for a periodic function of period a instead of 27 can be written down from the above form by merely changing @ into 27m/a, the variable now being x. Any finite periodic function, subject to certain limitationg of little practical moment, can be exe pre ed by means of a Fourier series, This very important result is known as Fourier’s Theorem. It was proved and applied to problems of heat cons duction by Fourier in his beautiful work mentioned in the preceding article. The theorem has an application to functions which are not periodic, In fact, if we are given the values of a function of x for all values of x between two end values r=0 and r=@ say, we may regard these data as defining one wave of a periodic function of x, the period of which is a; in other words, we can define a periodic function so that one wave of it may have any a igned form. Hence any ordinary function whatever can be represented by a Fourier series for all values of a between two a igned end values. This remarkable property was a_ great stumbling-block to early investigators who looked on such an equation ag 4x=sinx—}sin2x+4sin3x—fsindr+... as obviously erroneous, the right hand member being periodic, while the left is not. The result, however, is quite correct for values of « between—7z and 7, exclusive of these end values. A rigorous proof, or even statement, of Fourier’s theorem in all its po ible generality is not easy, the difficulty being chiefly due to the very great complexity of the modern conception of a function. On the other hand, if the theorem is a umed to be true, it is usually an easy matter to determine the coefficients in the expansion of a given function. Suppose, e.g., we require a series equivalent to the function x between x=—randz=r. We put L=Ay + a, COSY + Aq cos2r +... +b, sing +b, sinde+... To find 6», multiply both sides by sinnzx, and then integrate from « = — 7 to x = 7, a uming this to be allowable, We get. wv wv | x sinnx dx = bn sin’nx dx3 C) VAC for all the other integrals on the right are easily proved to disappear. Thus bn = — 2cosn7/n; similarly an = 0, and the series is found in the form already written down above. In almost every branch of applied mathematics, Fourier series are indispensable. As an example of their use, we may take the problem of a vibrating string, in the treatment of which by Daniel Bernouilli the series first made their appearance in analysis. It is known from the dynamics of the problem that a string held in the form y=sinnz (where y is the displacement of a particle at distance x from one 536 FOURMIES end), and let go at the time ¢=0, subject to the end points x= 0 and x=7 remaining fixed, will vibrate so that the displacement at time ¢ is; y=sinnxz cosnVt, | V being a constant depending on the tension and density of the string. Now take the corresponding problem for a string with the same fixed ends, but with any initial form. We can expand the value of y for this initial form in the series y=a, sintx+a, sm2r+.... The value of y at any time ¢ is then f y=a, sinx cosVi+d, sin2x cos, Vi+... The purely mathematical resolution of the original displacement into sinusoidal components a, sinx, d, sin2x, ., thus corresponds closely to that physical resolution of the vibration into harmonic components, which seems to be carried out by the ear in the proce of hearing.—BIBLIoGRAPHY: H.S. Carslaw, Fourier Series and Integrals; W. E. Byerly, Fourier’s Series and Spherical Harmonics; E. W. Hobson, Functions of a Real Variable and Fourier’s Series; papers by G. A. Gibson, Proceedings of the Edinburgh Mathematical Society, vols. ll and 12.