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CSS Applied Mathematics QUESTION #1314
Question 1
The Fourier series of a function \(f(x)\) on \((-\pi,\pi)\) is given by \(f(x)=\dfrac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\). The coefficient \(a_0\) is:
  • \(\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx\)✔️
  • \(\dfrac{2}{\pi}\int_{-\pi}^{\pi}f(x)\,dx\)
  • \(\int_{-\pi}^{\pi}f(x)\,dx\)
  • \(\dfrac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx\)
Correct Answer Explanation
\(a_0=\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx\). Then \(a_n=\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx\,dx\) and \(b_n=\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin nx\,dx\). The constant term in the series is \(\dfrac{a_0}{2}\).