Find the Fourier series representation of the function with period 2π defined by

f (t) = t² ,0 < t ≤ 2π.

Question

Find the Fourier series representation of the function with period 2π defined by

f (t) = t² ,0 < t ≤ 2π.

Accepted Answer

As usual we sketch f (t), as shown in Figure 23.16. Here T = 2π and we shall integrate, for convenience, over the interval [0, 2π]. Using Equation (23.3) we find

[latex]a_0=\frac{1}{\pi} \int_0^{2 \pi} t^2 \mathrm{~d} t=\frac{1}{\pi}\left[\frac{t^3}{3} ight]_0^{2 \pi}=\frac{8 \pi^2}{3}[/latex]

Using Equation (23.4) we have
[latex]a_n=\frac{2}{T} \int_0^T f(t) \cos \frac{2 n \pi t}{T} \mathrm{~d} t \quad ext { for } n ext { a positive integer }[/latex] (23.4)

[latex]a_n=\frac{1}{\pi} \int_0^{2 \pi} t^2 \cos n t \mathrm{~d} t[/latex]

Integrating by parts, we find

[latex]\begin{aligned} a_n & =\frac{1}{\pi}\left(\left[t^2 \frac{\sin n t}{n} ight]_0^{2 \pi}-\int_0^{2 \pi} 2 t \frac{\sin n t}{n} \mathrm{~d} t ight) \ & =-\frac{2}{n \pi} \int_0^{2 \pi} t \sin n t \mathrm{~d} t \ & =-\frac{2}{n \pi}\left(\left[-t \frac{\cos n t}{n} ight]_0^{2 \pi}+\int_0^{2 \pi} \frac{\cos n t}{n} \mathrm{~d} t ight) \ & =-\frac{2}{n \pi}\left(\frac{-2 \pi \cos 2 n \pi}{n}+\left[\frac{\sin n t}{n^2} ight]_0^{2 \pi} ight) \ & =\frac{4}{n^2} \end{aligned}[/latex]

Hence [latex]a_1=4, a_2=1, a_3=\frac{4}{9}, \ldots[/latex] Similarly,

[latex]\begin{aligned} b_n & =\frac{1}{\pi} \int_0^{2 \pi} t^2 \sin n t \mathrm{~d} t \ & =\frac{1}{\pi}\left(\left[-t^2 \frac{\cos n t}{n} ight]_0^{2 \pi}+\int_0^{2 \pi} 2 t \frac{\cos n t}{n} \mathrm{~d} t ight) \ & =\frac{1}{\pi}\left(-\frac{4 \pi^2}{n} \cos 2 n \pi+\frac{2}{n} \int_0^{2 \pi} t \cos n t \mathrm{~d} t ight) \ & =\frac{1}{\pi}\left(-\frac{4 \pi^2}{n}+\frac{2}{n}\left(\left[\frac{t \sin n t}{n} ight]_0^{2 \pi}-\int_0^{2 \pi} \frac{\sin n t}{n} \mathrm{~d} t ight) ight) \ & =\frac{1}{\pi}\left(-\frac{4 \pi^2}{n}-\frac{2}{n^2}\left[-\frac{\cos n t}{n} ight]_0^{2 \pi} ight) \ & =-\frac{4 \pi}{n} \end{aligned}[/latex]

Thus [latex]b_1=-4 \pi, b_2=-2 \pi, \ldots .[/latex] Finally, the required Fourier series representation is given by

[latex]\begin{aligned} f(t)= & \frac{4 \pi^2}{3}+\left(4 \cos t+\cos 2 t+\frac{4}{9} \cos 3 t+\cdots ight) \ & -\pi\left(4 \sin t+2 \sin 2 t+\frac{4 \sin 3 t}{3}+\cdots ight) \end{aligned}[/latex]

Question 23.15: Find the Fourier series representation of the function with ......

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