Page 346 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 13] FOURIER SERIES 337

FOURIER SERIES
Let f ðxÞ be defined in the interval ð L; LÞ and outside of this interval by f ðx þ 2LÞ¼ f ðxÞ, i.e., f ðxÞ
is 2L-periodic. It is through this avenue that a new function on an infinite set of real numbers is created
from the image on ð L; LÞ. The Fourier series or Fourier expansion corresponding to f ðxÞ is given by

1
X n x n x
a 0
a n cos þ b n sin ð1Þ
þ
2 L L
n¼1
where the Fourier coefficients a n and b n are
8 ð L
1 n x
>
> f ðxÞ cos dx
> a n ¼
< L
L L
n ¼ 0; 1; 2; ...
1 ð L n x ð2Þ
>
>
> f ðxÞ sin dx
:
b n ¼
L L L
ORTHOGONALITY CONDITIONS FOR THE SINE AND COSINE FUNCTIONS
Notice that the Fourier coefficients are integrals. These are obtained by starting with the series, (1),
and employing the following properties called orthogonality conditions:
ð L m x n x
(a) cos cos dx ¼ 0if m 6¼ n and L if m ¼ n
L L L
L m x n x
ð
(b) sin sin dx ¼ 0if m 6¼ n and L if m ¼ n (3)
L L L
L m x n x
ð
(c) sin cos dx ¼ 0. Where m and n can assume any positive integer values.
L L L
An explanation for calling these orthogonality conditions is given on Page 342. Their application in
determining the Fourier coefficients is illustrated in the following pair of examples and then demon-
strated in detail in Problem 13.4.
EXAMPLE 1. To determine the Fourier coefficient a 0 ,integrate both sides of the Fourier series (1), i.e.,
L L L X n n x n x o
ð ð ð
1
a 0
a n cos þ b n sin dx
f ðxÞ dx ¼ dx þ
L L 2 L n¼1 L L
ð L ð L ð L ð L
a 0 n x n x 1
Now dx ¼ a 0 L; sin dx ¼ 0; cos dx ¼ 0, therefore, a 0 ¼ f ðxÞ dx
L 2 l L L L L L
x
EXAMPLE 2. To determine a 1 , multiply both sides of (1)by cos and then integrate. Using the orthogonality
1 ð L x L
f ðxÞ cos dx. Now see Problem 13.4.
conditions (3) a and (3) c ,we obtain a 1 ¼ L
L L
If L ¼ , the series (1) and the coefficients (2)or(3) are particularly simple. The function in this
case has the period 2 .
DIRICHLET CONDITIONS
Suppose that
(1) f ðxÞ is defined except possibly at a finite number of points in ð L; LÞ
(2) f ðxÞ is periodic outside ð L; LÞ with period 2L