Richard Bronson - Schaums Outline of Differential Equations, 3rd edition

A wide class of functions can be represented by infinite series of eigenfunctions of a SturmLiouville problem (see ). Definition: A function f(x) is piecewise continuous on the open interval a <x <b if (1) f(x) is continuous everywhere in a <x <b with the possible exception of at most a finite number of points x, x, , x n and (2) at these points of discontinuity, the right- and left-hand limits of f(x), respectively lim f(x) and lim f(x), exist (j = 1, 2,..., n). (Note that a continuous function is piecewise continuous.) Definition: A function f(x) is piecewise continuous on the closed interval axb if (1) it is piecewise continuous on the open interval a <x <b, (2) the right-hand limit of f(x) exists at x = a, and (3) the left-hand limit of f(x) exists at x = b. Definition: A function f(x) is piecewise smooth on [a, b] if both f(x) and f(x) are piecewise continuous on [a, b]. Theorem 33.1. If f(x) is piecewise smooth on [a, b] and if {e n (x)} is the set of all eigenfunctions of a SturmLiouville problem (see Property 32.3), then where The representation (33.1) is valid at all points in the open interval (a, b) where f(x) is continuous. (32.6). (32.6).

Because different SturmLiouville problems usually generate different sets of eigenfunctions, a given piecewise smooth function will have many expansions of the form (33.1). The basic features of all such expansions are exhibited by the trigonometric series discussed below.

The eigenfunctions of the SturmLiouville problem y" + y = 0; y(0) = 0, y(L) = 0, where L is a real positive number, are e n (x) = sin (nx/L) (n = 1, 2, 3, ). Substituting these functions into (33.1), we obtain For this SturmLiouville problem, w(x) 1, a = 0, and b = L; so that and (33.2) becomes The expansion (33.3) with coefficients given by (33.4) is the Fourier sine series for f(x) on (0, L). The eigenfunctions of the SturmLiouville problem y" + y = 0; y(0) = 0, y(L) = 0, where L is a real positive number, are e(x) = 1 and e n (x) = cos (nx/L) (n = 1, 2, 3, ). Here = 0 is an eigenvalue with corresponding eigenfunction e(x) = 1.

Substituting these functions into (33.1), where because of the additional eigenfunction e(x) the summation now begins at n = 0, we obtain For this SturmLiouville problem, w(x) 1, a = 0, and b = L; so that Thus (33.2) becomes The expansion (33.5) with coefficients given by (33.6) is the Fourier cosine series for f(x) on (0, L).

33.1. Determine whether is piecewise continuous on [1, 1]. The given function is continuous everywhere on [1, 1] except at x = 0. Therefore, if the right- and left-hand limits exist at x = 0, f(x) will be piecewise continuous on [1, 1]. We have Since the left-hand limit does not exist, f(x) is not piecewise continuous on [1, 1]. (Note that f(x) is continuous at x = 1.) At the two points of discontinuity, we find that and Since all required limits exist, f(x) is piecewise continuous on [2, 5]. 33.3. Is the function piecewise smooth on [2, 2]? The function is continuous everywhere on [2, 2] except at x = 1. 33.3. Is the function piecewise smooth on [2, 2]? The function is continuous everywhere on [2, 2] except at x = 1.

Since the required limits exist at x, f(x) is piecewise continuous. Differentiating f(x), we obtain The derivative does not exist at x = 1 but is continuous at all other points in [2, 2]. At x the required limits exist; hence f(x) is piecewise continuous. It follows that f(x) is piecewise smooth on [2, 2]. 33.4. Is the function piecewise smooth on [1, 3]? The function f(x) is continuous everywhere on [1, 3] except at x = 0. Since the required limits exist at