Question

Which of the following statements about Fourier series is/are correct?

• A. The Fourier series of an even function contains only cosine terms
• B. The Fourier series of an odd function contains only cosine terms
• C. The Fourier series of an odd function contains only sine terms
• D. The Fourier series of an even function contains only sine terms

Hint:

Work out whether \(f(x)\cos(n\omega x)\) and \(f(x)\sin(n\omega x)\) are even or odd for an even \(f(x)\) and for an odd \(f(x)\), then recall that an odd function integrates to zero over a symmetric interval.

Answer 1

The Correct Option is A, C

Step 1: Understanding the Question:
A Fourier series writes a periodic function as a sum of a constant term plus cosine and sine terms:
\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(n\omega x) + b_n \sin(n\omega x) \right) \]
We need to check what happens to the coefficients \(a_n\) and \(b_n\) when \(f(x)\) is even or odd, since that decides whether the series keeps only cosine terms, only sine terms, or both.

Step 2: Key Formula or Approach:
Over a symmetric interval \([-L, L]\), the coefficients are
\[ a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos(n\omega x)\,dx, \qquad b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin(n\omega x)\,dx \]
The integral of an odd function over a symmetric interval is always zero, while the integral of an even function over a symmetric interval is twice the integral over the positive half. We use this together with the rules: even times even is even, odd times odd is even, and even times odd is odd.

Step 3: Detailed Explanation:
Case 1, \(f(x)\) even: \(\cos(n\omega x)\) is always even, so \(f(x)\cos(n\omega x)\) is even times even, which is even, and its integral over \([-L,L]\) is generally nonzero, so \(a_n\) survives. But \(\sin(n\omega x)\) is odd, so \(f(x)\sin(n\omega x)\) is even times odd, which is odd, and its integral over the symmetric interval is zero. So \(b_n = 0\) for every \(n\), and the series of an even function keeps only the constant term and cosine terms. This matches option (A) and rules out option (D).
Case 2, \(f(x)\) odd: now \(f(x)\cos(n\omega x)\) is odd times even, which is odd, so its integral is zero and \(a_n = 0\) for all \(n\), including \(a_0\). But \(f(x)\sin(n\omega x)\) is odd times odd, which is even, so its integral survives and \(b_n\) is generally nonzero. So the series of an odd function keeps only sine terms. This matches option (C) and rules out option (B).

Step 4: Final Answer:
An even function's Fourier series has only cosine terms, and an odd function's Fourier series has only sine terms. So statements (A) and (C) are correct, while (B) and (D) are the reversed, incorrect versions of these facts.
\[ \boxed{\text{A and C}} \]