Page 367 - Schaum's Outline of Theory and Problems of Advanced Calculus

358 FOURIER SERIES [CHAP. 13

! 2 ! 2
1
ð L r ffiffiffiffi 1 m x ð L r ffiffiffiffi m x
Then sin dx ¼ 1; cos dx ¼ 1
L L L L L L
L L 1
ð ð 2
2
Also, ð1Þ dx ¼ 2L or p ffiffiffiffiffiffi dx ¼ 1
L L 2L
Thus the required orthonormal set is given by
1 1 x 1 x 1 2 x 1 2 x
ffiffiffiffiffiffi ; p sin ffiffiffiffi cos ; p sin ffiffiffiffi cos ; ...
p ffiffiffiffi ; p ffiffiffiffi ; p
2L L L L L L L L L

MISCELLANEOUS PROBLEMS

13.26. Find a Fourier series for f ðxÞ¼ cos x; @ x @ , where 6¼ 0; 1; 2; 3; ... .
We shall take the period as 2 so that 2L ¼ 2 ; L ¼ . Since the function is even, b n ¼ 0 and
2 ð L 2 ð
cos x cos nx dx
a n ¼ f ðxÞ cos nx dx ¼
L 0 0
1 ð
fcosð nÞx þ cosð þ nÞxg dx
¼
0
1 sinð nÞ sinð þ nÞ 2 sin cos n

n þ n ð n Þ
¼ þ ¼ 2 2
2 sin

0 ¼
Then
sin 2 sin X cos n
1
cos nx
n
cos x ¼ þ 2 2
n¼1

sin 1 2 2 2
1 2 3
¼ 2 2 cos x þ 2 2 cos 2x 2 2 cos 3x þ
! ! !
2 2 2
x x x
.
13.27. Prove that sin x ¼ x 1 2 1 2 1 2

ð2 Þ ð3 Þ
Let x ¼ in the Fourier series obtained in Problem 13.26. Then

sin 1 2 2 2
1 2 3
cos ¼ þ 2 2 þ 2 2 þ 2 2 þ
or
1 2 2 2
1 2 3
cot ¼ 2 2 þ 2 2 þ 2 2 þ ð1Þ
This result is of interest since it represents an expansion of the contangent into partial fractions.
By the Weierstrass M test, the series on the right of (1)converges uniformly for 0 @ j j @ jxj < 1 and
the left-hand side of (1) approaches zero as ! 0, as is seen by using L’Hospital’s rule. Thus, we can
integrate both sides of (1) from 0 to x to obtain
x ð x ð x
ð
1 2 2
cot d ¼ 2 d þ 2 2 d þ
0 0 1 0 2
! !
x 2 2
sin x x
or ln ¼ ln 1 þ ln 1 þ
0 1 2 2 2