Page 368 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 13] FOURIER SERIES 359

! ! !
2 2 2
sin x x x x
i.e., ln ¼ lim ln 1 þ ln 1 þ þ ln 1
x n!1 1 2 2 2 n 2
( 2 ! 2 ! 2 !)
x x x
¼ lim ln 1 1 1
1 2 2 2 n 2
n!1
( 2 ! 2 ! 2 !)
x x x
¼ ln lim 1 1 1
1 2 2 2 n 2
n!1
so that
! ! ! ! !
sin x x 2 x 2 x 2 x 2 x 2
¼ lim
x n!1 1 1 2 1 2 2 1 n 2 ¼ 1 1 2 1 2 2 ð2Þ
Replacing x by x= ,we obtain
! !
x 2 x 2
ð2 Þ
sin x ¼ x 1 2 1 2 ð3Þ
called the infinite product for sin x, which can be shown valid for all x. The result is of interest since it
corresponds to a factorization of sin x in a manner analogous to factorization of a polynomial.
2 2 4 4 6 6 8 8 ...
13.28. Prove that ¼ .
2 1 3 3 5 5 7 7 9 ...
Let x ¼ 1=2in equation (2)ofProblem 13.27. Then,

2 1 1 1 1 3 3 5 5 7
¼ 1 2 2 1 4 2 1 6 2 ¼ 2 2 4 4 6 6

Taking reciprocals of both sides, we obtain the required result, which is often called Wallis’ product.

Supplementary Problems

FOURIER SERIES
13.29. Graph each of the following functions and find their corresponding Fourier series using properties of even
and odd functions wherever applicable.
80 < x < 2 x 4 @ x @ 0

Period 4 Period 8
82 < x < 4 x 0 @ x @ 4
ðaÞ f ðxÞ¼ ðbÞ f ðxÞ¼
2x 0 @ x < 3

ðcÞ f ðxÞ¼ 4x; 0 < x < 10; Period 10 ðdÞ f ðxÞ¼ Period 6
0 3 < x < 0
16 X n x 8 X n x
1
1
Ans: ðaÞ ð1 cos n Þ sin ðbÞ 2 ð1 cos n Þ cos
n 2 2 n 2 4
n¼1 n¼1

40 X 1 n x 3 X 6ðcos n 1Þ n x 6cos n n x
1
1
n 5 2 n 3 n 3
ðcÞ 20 sin ðdÞ þ 2 2 cos sin
n¼1 n¼1
13.30. In each part of Problem 13.29, tell where the discontinuities of f ðxÞ are located and to what value the series
converges at the discontunities.
Ans. (a) x ¼ 0; 2; 4; ...; 0 ðbÞ no discontinuities (c) x ¼ 0; 10; 20; .. .; 20
(d) x ¼ 3; 9; 15; .. .; 3