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GATE 2025 CE (CE1) – Question 13

Engineering Mathematics · PDE: Fourier series, separation of variables, diffusion, wave and Laplace equations · 1 mark · Multiple choice

Which one of the following options is the correct Fourier series of the periodic function $f(x)$ described below:

$$f(x) = \begin{cases} 0, & -2 < x < -1 \\ 2k, & -1 < x < 1 \\ 0, & 1 < x < 2 \end{cases}; \qquad \text{period} = 4$$

  1. $f(x) = \frac{k}{2} + \frac{2k}{\pi}\left(\cos\frac{\pi}{2}x - \frac{1}{3}\cos\frac{3\pi}{2}x + \frac{1}{5}\cos\frac{5\pi}{2}x - + \cdots\right)$
  2. $f(x) = \frac{k}{2} + \frac{2k}{\pi}\left(\sin\frac{\pi}{2}x - \frac{1}{3}\sin\frac{3\pi}{2}x + \frac{1}{5}\sin\frac{5\pi}{2}x - + \cdots\right)$
  3. $f(x) = k + \frac{4k}{\pi}\left(\cos\frac{\pi}{2}x - \frac{1}{3}\cos\frac{3\pi}{2}x + \frac{1}{5}\cos\frac{5\pi}{2}x - + \cdots\right)$
  4. $f(x) = k + \frac{4k}{\pi}\left(\sin\frac{\pi}{2}x - \frac{1}{3}\sin\frac{3\pi}{2}x + \frac{1}{5}\sin\frac{5\pi}{2}x - + \cdots\right)$

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Correct answer: (C) $f(x) = k + \frac{4k}{\pi}\left(\cos\frac{\pi}{2}x - \frac{1}{3}\cos\frac{3\pi}{2}x + \frac{1}{5}\cos\frac{5\pi}{2}x - + \cdots\right)$

Explanation

The function is even, so only cosine terms appear. The mean value is $\frac{1}{4}\int_{-1}^{1} 2k\,dx = k$. The coefficients are $a_n = \frac{2}{4}\int_{-1}^{1} 2k\cos\frac{n\pi x}{2}\,dx = \frac{4k}{n\pi}\sin\frac{n\pi}{2}$, which is $\frac{4k}{\pi}$, $-\frac{4k}{3\pi}$, $\frac{4k}{5\pi}$ for $n = 1, 3, 5$ and zero for even $n$. So $f(x) = k + \frac{4k}{\pi}\left(\cos\frac{\pi x}{2} - \frac{1}{3}\cos\frac{3\pi x}{2} + \frac{1}{5}\cos\frac{5\pi x}{2} - \cdots\right)$.