Fourier Series and Partial Differential Equations Lecture Notes

Chapter 2. Fourier series 23
Note. For fo to really be odd we must have fo(0) = 0 and also fo(L) = −fo(−L) =
−fo(L) (thelast equality follows fromperiodicity)givingfo(L) = 0andthereforefo(nL) =
0 for all n ∈ Z. However, the value at of f at these isolated points does not affect the
Fourier series.
Example 2.4 Find the Fourier sine and cosine expansions of f(x) = x for x ∈ [0,L].
Sine expansion The odd extension is defined by
In this case
fo(x) =
0
x x ∈ [0,L],
−(−x) x ∈ (−L,0).
−L
y
2L
x
(2.83)
bn = 2
L
nπx
xsin dx = (−1)
L L
n+12L
, (2.84)
nπ
as in Example 2.3. For x ∈ [0,L) we therefore obtain
x =
∞
(−1) n+12L
nπ sin
nπx
. (2.85)
L
n=1
Cosine expansion The even extension is given by
Now,
and
Thus for x ∈ [0,L] we get
fe(x) =
x x ∈ [0,L],
−L
a0 = 2
L
−x x ∈ [−L,0).
y
2L
x
(2.86)
L
xdx = L, (2.87)
0
an = 2
⎧
L
nπx
⎨0
n = 2m even,
xcos dx =
L 0 L ⎩ −4L
(2m+1) 2π2 n = 2m+1 odd.
x = L
2 +
m=0
(2.88)
∞ 4L
−
(2m+1) 2
(2m+1)πx
cos . (2.89)
π2 L