434 Chapter 15Orthogonal expansions. Fourier analysis
The infinite interval
The Fourier series (15.48) for arbitrary interval 2 l,
(15.61)
can be written as
(15.62)
where
A sum like (15.62) can be considered as the sum of
the areas of rectangles of width∆n 1 = 11 and height
c
n, as illustrated in Figure 15.12,
(15.63)
We now make the the substitutions
(15.64)
Equation (15.61) is then
(15.65)
where, by (15.49) and (15.50) for the Fourier coefficients,
(15.66)
(15.67)
We are now in a position to letl 1 → 1 ∞and∆y
n1 → 10. Equation (15.65) has the form
(15.68)
fx Fy ynnn()=∆( )
=∑
0∞v()y ()sin
lb
fx xydx
nnlln==
−+ππ
1
Z
uy
la
fx xydx
nnlln()== ()cos
−+ππ
1
Z
fx
uy
uy xy y xy
nnnnn()
()
=+()cos ()sin+
=∑
012
∞v
∆y
ny
l
ny
l
nuy
l
ay
l
b
nn nnnn=,∆=∆ = =
ππ
ππ
,() ,()v
fx c n
nn()=∆
=∑
0∞c
a
ca
nx
l
b
nx
l
n
0 nn n02
=, =cos +sin , > 0
ππ
fx c
nn()=
=∑
0∞fx
a
a
nx
l
b
nx
l
nnn()=+ cos +sin
=∑
012
∞ππ
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...
..
..
...
..
...
......
....
....
.....
....
....
...0 1 234
···
c ···
0c
1c
2c
3n
f(n)
Figure 15.12