Page 348 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 13] FOURIER SERIES 339

DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES
Differentiation and integration of Fourier series can be justified by using the theorems on Pages 271
and 272, which hold for series in general. It must be emphasized, however, that those theorems provide
sufficient conditions and are not necessary. The following theorem for integration is especially useful.

Theorem. The Fourier series corresponding to f ðxÞ may be integrated term by term from a to x, and the
ð x
resulting series will converge uniformly to f ðxÞ dx provided that f ðxÞ is piecewise continuous in
a
L @ x @ L and both a and x are in this interval.

COMPLEX NOTATION FOR FOURIER SERIES
Using Euler’s identities,
i
e ¼ cos þ i sin ; e i ¼ cos i sin ð6Þ
p ffiffiffiffiffiffiffi
1 (see Problem 11.48, Chapter 11, Page 295), the Fourier series for f ðxÞ can be written as
where i ¼
1
X in x=L
c n e
f ðxÞ¼ ð7Þ
n¼ 1
where
1 ð L in x=L
f ðxÞe dx
c n ¼ ð8Þ
2L L
In writing the equality (7), we are supposing that the Dirichlet conditions are satisfied and further
that f ðxÞ is continuous at x. If f ðxÞ is discontinuous at x, the left side of (7)should be replaced by
:
ðf ðx þ 0Þþ f ðx 0Þ
2

BOUNDARY-VALUE PROBLEMS
Boundary-value problems seek to determine solutions of partial differential equations satisfying
certain prescribed conditions called boundary conditions. Some of these problems can be solved by
use of Fourier series (see Problem 13.24).

EXAMPLE. The classical problem of a vibrating string may be idealized in the following way. See Fig. 13-2.
Suppose a string is tautly stretched between points ð0; 0Þ and ðL; 0Þ. Suppose the tension, F,is the
same at every point of the string. The string is made to
vibrate in the xy plane by pulling it to the parabolic
2
position gðxÞ¼ mðLx x Þ and releasing it. (m is a
numerically small positive constant.) Its equation will
be of the form y ¼ f ðx; tÞ. The problem of establishing
this equation is idealized by (a) assuming that the con-
stant tension, F,isso large as compared to the weight wL
of the string that the gravitational force can be neglected,
(b) the displacement at any point of the string is so small
that the length of the string may be taken as L for any of
its positions, and (c) the vibrations are purely transverse.
2
w @ y
The force acting on a segment PQ is x ;
g @t 2
2
x < x 1 < x þ x; g 32 ft per sec: . If and are the
angles that F makes with the horizontal, then the vertical Fig. 13-2