Page 336 - Schaum's Outline of Differential Equations

CHAP. 33] EIGENFUNCTION EXPANSIONS 319

FOURIER SINE SERIES
The eigenfunctions of the Sturm-Liouville problem /' + ky = 0; y(0) = 0, y(L) = 0, where L is a real positive
number, are e n(x) = sin (mccIL) (n = 1, 2, 3, ...). Substituting these functions into (33.1), we obtain

For this Sturm-Liouville 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/(;t) on (0, L).

FOURIER COSINE SERIES
The eigenfunctions of the Sturm-Liouville problem y" + A,;y = 0; /(O) = 0, y'(L) = 0, where L is a real positive
number, are e 0(x) = 1 and e n(x) = cos (njrxIL) (n = 1, 2, 3, ...). Here ^, = 0 is an eigenvalue with corresponding
eigenfunction e 0(x) = 1. Substituting these functions into (33.7), where because of the additional eigenfunction
e 0(x) the summation now begins at n = 0, we obtain

For this Sturm-Liouville 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).

Solved Problems

f
33.1. Determine whether ( x ) = 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].