312Differential Equations 2301312: ISE Program, Chulalongkorn Univer- sity: First semester
Exercises for Fourier Series
Find Fourier series for For each of the given functions. Assume that the functions are periodically extended outside the original interval.
(1.) f(x) =
(−1, −1≤x <0 1, 0≤x <1 (2.) f(x) =
(0, −π≤x <0 1, 0≤x < π (3.) f(x) = sin2x, −π≤x≤π (4.) f(x) = 1−x2, −1≤x≤1 (5.) f(x) =
(0, −1≤x <0 x2, 0≤x <1
Example: For (1.), function f(x) is defined for −1 ≤ x < 1. We can extend the function periodically outside this interval with period T = 2 by setting f(x+ 2) =f(x).
The functionf(x) thus has the Fourier series, (L= 1), f(x) = a0
2 +
∞
X
n=1
ancosnπx+bnsinnπx.
a0 = Z 1
−1
f(x)dx= Z 0
−1
(−1)dx+ Z 1
0
1dx=−1 + 1 = 0
an= Z 1
−1
f(x) cosnπx dx= Z 0
−1
−cosnπx dx+ Z 1
0
cosnπx dx
=
·
− 1
nπ sinnπx
¸0
−1
+
· 1
nπ sinnπx
¸1 0
= 1
nπ (−sinnπ+ sinnπ) an= 0 (n= 1,2,3, . . .)
bn= Z 1
−1
f(x) sinnπx dx= Z 0
−1
−sinnπx dx+ Z 1
0
sinnπx dx
=
· 1
nπ cosnπx
¸0
−1
+
·
− 1
nπ cosnπx
¸1 0
= 1
nπ (1−cosnπ−cosnπ+ 1) = 2
nπ (1−cosnπ)
∴ bn = 2 nπ
³
1−cosnπ 2
´
(n = 1,2,3, . . .) The Fourier series for f(x) is
∞
X
n=1
2
nπ (1−cosnπ) sinnπx.
Fourier Cosine and Sine Series 1
Expanding a function f defined on [0, L] in Fourier Series.
1. Fourier Cosine Series. We extend f to be an even periodic function with period 2Lusing
g(x) :=
(f(x), 0≤x≤L, f(−x), −L < x <0.
2. Fourier Sine Series. We extend f to be an odd periodic function with period 2Lusing
h(x) :=
f(x), 0< x < L,
0, x= 0, L
−f(−x), −L < x < 0.
3. Fourier Series (Normal Form). We extendf to be a periodic function with period L using f(x+L) =f(x).
Exercises Find the Fourier cosine and sine series forf: (1.) f(x) =
(1, 0< x <1, 0, 1< x <2 (2.) f(x) =x, 0≤x <1 (3.) f(x) =
(x, 0< x < π, 0, π < x <2π (4.) From the given Fourier series of a periodic function f(x)
f(x) = L
2 +−4L π2
∞
X
n=1
cos(2n−L1)πx (2n−1)2 , f(x) =
(−x, −L≤x <0,
x, 0≤x < L; f(x+ 2L) =f(x).
Show that
π2
8 = 1 + 1 32 + 1
52 +· · ·=
∞
X
n=1
1 (2n+ 1)2.
2
