Mathematical Methods for Physicists: A concise introduction - Site Map

ALTERNATIVE FORMS OF FOURIER SERIES
then the average
a 2 0
2 ‡ X1
nˆ1
a 2 n ‡ b 2 …2† 2
n ˆ
2 ‡ X1
nˆ1
16
n 4 cos n 1
4 … †2 :
Parseval's identity now becomes
8
3 ˆ 2 ‡ 64
1
4 1 4 ‡ 1 3 4 ‡ 1
5 4 ‡ ;
or
1
1 4 ‡ 1 3 4 ‡ 1 4
‡ˆ
54 96
which shows that we can use Parseval's identity to ®nd the sum of an in®nite
series. With the help of the above result, we can ®nd the sum S of the following
series:
1
1 4 ‡ 1 2 4 ‡ 1 3 4 ‡ 1 4 4 ‡‡ 1 n 4 ‡:
S ˆ 1
1 4 ‡ 1 2 4 ‡ 1 3 4 ‡ 1
4 4 ‡ˆ 1
1 4 ‡ 1 3 4 ‡ 1
5 4 ‡ ‡ 1 2 4 ‡ 1 4 4 ‡ 1
6 4 ‡
ˆ 1
1 4 ‡ 1 3 4 ‡ 1
5 4 ‡ ‡ 1
1
2 4 1 4 ‡ 1 2 4 ‡ 1 3 4 ‡ 1
4 4 ‡
ˆ 4
96 ‡ S 16
from which we ®nd S ˆ 4 =90.
Alternative forms of Fourier series
Up to this point the Fourier series of a function has been written as an in®nite
series of sines and cosines, Eq. (4.2):
f …x† ˆa0
2 ‡ X1
nˆ1
a n cos nx
L ‡ b n sin nx
:
L
This can be converted into other forms. In this section, we just discuss two alternative
forms. Let us ®rst write, with =L ˆ
q
!
a n cos nx ‡ b n sin nx ˆ a 2 n ‡ b 2 a n
b
n p cos nx ‡ p n
sin nx :
a 2 n ‡ b 2 n
a 2 n ‡ b 2 n
155