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and
Z
1
x
2
sin.kx/
dx
D
x
2
k
cos.kx/
1
0
Z
1
0
1
1
k
cos.kx/
dx
2x
0
x
k
sin.kx/
1
0
D
1
2
k
1
k
cos.k/
C
Z
1
2
k
1
k
sin.kx/
dx
0
1
k
cos.kx/
1
0
D
1
2
.k/
2
k
cos.k/
C
D
1
2
.k/
3
2
.k/
3
k
cos.k/
C
cos.k/
.
1/
k
C
1
k
2.
1/
k
.k/
3
2
.k/
3
D
C
:
Thus
D
2
2
.k/
3
!
D
(
0
2.
1/
k
.k/
3
if k is even;
c
k
8
.k/
3
if k is odd;
and, at least formally,
25
we find that
sin..2k
1/x/:
X
8
..2k
1//
3
f.x/
D
x
x
2
D
k
D
1
In Fig.
8.4
we have plotted the functions defined by the first, the first seven, and
the first 100 terms of this series. Note that
f.0/
D
f.1/
D
0;
and consequently it seems that f is accurately approximated by as few as seven
terms of its Fourier series, see the discussion following Example
8.5
.
Example 8.8.
We want to solve the problem
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
u
.0; t /
D
u
.1; t /
D
0 for t>0;
u
.x; 0/
D
x.1
x/
D
x
x
2
for x
2
.0; 1/:
By the formula derived in the previous example we find that this problem can be
written in the form
25
We have not proved that the series converges toward the correct limit!
