MAT3700 Assignment 04 (Accurate Solutions) 2025 Due Year 2025

MAT3700
Assignment 04

Accurate Solutions

Year 2025

, MAT3700 Assignment 04 — Due: Year 2025




MAT3700 Assignment 04
Due Date:Year 2025


Question 1

Problem Statement

A periodic function f (t) with period 2π is defined by


f (t) = t2 + t, (−π < t < π), f (t) = f (t + 2π).


Obtain the Fourier series expansion of f (t).


Step 1: Fourier Series Setup

For a 2π-periodic function,

∞
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: Compute a0

Since t2 is even and t is odd,

π π
2 π3 2π 2
Z Z
1 2 2
a0 = (t + t) dt = t2 dt = · = .
π −π π 0 π 3 3

Thus, the constant term is
a0 π2
= .
2 3

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