TC Asgn5
TC Asgn5
TC Asgn5
Transform Calculus
(MA20202)
Assignment-5
in cosine series.
3. Expand the function f (x), defined by
(
x, 0 ≤ x ≤ 2l
f (x) =
l − x, 2l < x ≤ l
in Fourier series.
1
Spring 2023-2024
π2
cos 2x cos 4x cos 6x
a. x(π−x) = − + + + . . .
6 12 22 32
8 sin x sin 3x sin 5x
b. x(π−x) = − + + + ...
π 12 32 52
Deduce form a. and b., respectively, that
∞
X 1 π4
c. =
n=1
n4 90
∞
X 1 π4
d. =
n=1
n6 945
8. Given that f (x) = x+x2 for −π < x < π, find the Fourier series expression
of f (x). Deduce that
π2 1 1 1
= 1 + 2 + 2 + 2 + ...
6 2 3 4
x2
9. Find the Fourier series expansion for f (x) = x + 4 , −π ≤ x ≤ π.
10. Expand the function f (x) = x sin x, as a Fourier series in the interval
−π ≤ x ≤ π. Hence deduce that
1 1 1 1 π−2
− + − + ... = .
1.3 3.5 5.7 7.9 4
12. Find the Fourier series corresponding to the function f (x) defined in
(−2, 2) as follows: (
2, −2 < x ≤ 0
f (x) =
x, 0 < x < 2