Fourier Series for first year Engineering Mathematics

Fourier Series
1) Obtain the Fourier series of 𝑓(𝑥) = 𝑥 − 𝑥2 in (−𝜋, 𝜋). Hence deduce that

𝜋 2 1 1 1 1


2) Obtain Fourier series for 𝑓(𝑥) = |𝑥| over the interval −π ≤ x ≤ π. Hence deduce that


3) Obtain the Fourier series for the function 𝑥2 in the interval(−𝜋, 𝜋). Hence deduce that
2
𝜋 1
= − 2
+−… … …
12 1

4) Obtain the Fourier series for the function 𝑓(𝑥) = 𝑒−𝑥 in the interval(0, 2𝜋).

5) Obtain the Fourier series for the function 𝑓(𝑥) = 𝑥(2𝜋 − 𝑥) in 0 ≤ 𝑥 ≤ 2𝜋. Hence deduce

𝜋
that =+ + 2
+ … … … ….
𝜋 −𝑥
8 6) Obtain the Fourier series for the function 𝑓(𝑥) = in the

interval (0, 2𝜋).
2

7) Obtain the Fourier series for the function 𝑓(𝑥) = 𝑥 in the interval(−3, 3).

8) Obtain the Fourier series for the function 𝑓(𝑥) = 2𝑥 − 𝑥2 in the interval (0, 3).

−𝜋 𝑖𝑛 −𝜋<𝑥<0
9) Obtain the Fourier series expansion for the function 𝑓(𝑥) = {
𝑥 𝑖𝑛 0<𝑥<𝜋
10) Obtain the Fourier series representation in (0, 2𝜋) of the function defined by

𝑥2 𝑖𝑛 (0, 𝜋)
𝑓(𝑥) = {
−(2𝜋 − 𝑥)2 𝑖𝑛 (𝜋, 2𝜋)
11) Obtain the Fourier series expansion for the function

𝑓(𝑥) = {1 + 2𝑥 𝑖𝑛 − 3 < 𝑥 < 0
1 − 2𝑥 𝑖𝑛 0<𝑥<3
12) Obtain the Fourier series representation in (0, 2) of the function defined by
𝜋𝑥 𝑖𝑛 (0, 1)