Page 335 - Schaum's Outline of Differential Equations

CHAPTER 33

Eigenfunction

Expansions

PIECEWISE SMOOTH FUNCTIONS

A wide class of functions can be represented by infinite series of eigenfunctions of a Sturm-Liouville problem
(see Chapter 32).

Definition: A function/(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 2, ... ,x n
and (2) at these points of discontinuity, the right- and left-hand limits of f(x), respectively
f
lim ( x ) and lim (x), exist (j= 1,2, ... , «).
f
(Note that a continuous function is piecewise continuous.)

Definition: A function/(X) is piecewise continuous on the closed interval a < x < b 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/^) is piecewise smooth on [a, b] if both/(j:) and/'(.*:) 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
Sturm-Liouville 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.
The function w(x) in (33.2) is given in Eq. (32.6).
Because different Sturm-Liouville problems usually generate different sets of eigenfunctions, a given
piecewise smooth function will have many expansions of the form (33.7). The basic features of all such expansions
are exhibited by the trigonometric series discussed below.

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