Page 339 - Schaum's Outline of Differential Equations

322 EIGENFUNCTION EXPANSIONS [CHAP. 33

Thus Eq. (33.3) becomes

Furthermore,

Hence,

Since/(X) is piecewise smooth on [0, 3] and continuous everywhere in (0, 3) except at x = 2, it follows from
Theorem 33.1 that (1) is valid everywhere in (0, 3) except at x = 2.

x
33.8. Find a Fourier sine series for/(X) = e on (0, n).
Using Eq. (33.4) with L = 7T, we obtain

Thus Eq. (33.3) becomes

It follows from Theorem 33.1 that this last equation is valid for all x in (0, it).

33.9. Find a Fourier cosine series for/(X) = e* on (0, n).
Using Eq. (33.6) with L = n, we have

Thus Eq. (33.5) becomes

As in Problem 33.8, this last equation is valid for all x in (0, n).