MAT3700
Assignment 04
Accurate Solutions
Year 2025
, Student Name: MAT3700 Assignment 04
MAT3700 Assignment 04
Due: Year 2025
Problem 1
Problem Statement
Find the full Fourier series of the 2π-periodic function
f (t) = t2 + t, −π < t < π, f (t + 2π) = f (t).
Step 1: Fourier series form
The Fourier series of a 2π-periodic function is
∞
a0 X
f (t) = + an cos(nt) + bn sin(nt) ,
2 n=1
where
Z π Z π Z π
1 1 1
a0 = f (t) dt, an = f (t) cos(nt) dt, bn = f (t) sin(nt) dt.
π −π π −π π −π
Step 2: Symmetry
Note:
t2 is even, t is odd.
Thus:
a0 : only t2 contributes, an : only t2 cos(nt) contributes, bn : only t sin(nt) contributes.
Step 3: Compute coefficients
π
2 π3 2π 2
Z
2
a0 = t2 dt = · = .
π 0 π 3 3
For an :
π
4(−1)n
Z
2
an = t2 cos(nt) dt = .
π 0 n2
Page 1
Assignment 04
Accurate Solutions
Year 2025
, Student Name: MAT3700 Assignment 04
MAT3700 Assignment 04
Due: Year 2025
Problem 1
Problem Statement
Find the full Fourier series of the 2π-periodic function
f (t) = t2 + t, −π < t < π, f (t + 2π) = f (t).
Step 1: Fourier series form
The Fourier series of a 2π-periodic function is
∞
a0 X
f (t) = + an cos(nt) + bn sin(nt) ,
2 n=1
where
Z π Z π Z π
1 1 1
a0 = f (t) dt, an = f (t) cos(nt) dt, bn = f (t) sin(nt) dt.
π −π π −π π −π
Step 2: Symmetry
Note:
t2 is even, t is odd.
Thus:
a0 : only t2 contributes, an : only t2 cos(nt) contributes, bn : only t sin(nt) contributes.
Step 3: Compute coefficients
π
2 π3 2π 2
Z
2
a0 = t2 dt = · = .
π 0 π 3 3
For an :
π
4(−1)n
Z
2
an = t2 cos(nt) dt = .
π 0 n2
Page 1
