Question

For an even function, the Fourier coefficients are:
A. \(a_0 \neq 0\)
B. \(a_n \neq 0\)
C. \(a_n = 0\)
D. \(b_n = 0\)
Choose the correct answer from the options given below:

• A. A, C and D only
• B. A, B and D only
• C. C and D only
• D. B and D only

Hint:

Remember the symmetry properties for Fourier series: - **Even function:** Contains only cosine terms (\(b_n = 0\)). - **Odd function:** Contains only sine terms (\(a_0 = 0\) and \(a_n = 0\)). This saves a lot of calculation time.

Answer 1

The Correct Option is B

Step 1: Define an even function and the Fourier coefficients. An even function satisfies \(f(-x) = f(x)\). The Fourier series is \(f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))\). - \(a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx\) - \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx) dx\) - \(b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx) dx\)
Step 2: Analyze the coefficients for an even function. - \(a_0\) and \(a_n\): The cosine function, \(\cos(nx)\), is also an even function. The product of two even functions (\(f(x)\) and \(\cos(nx)\)) is even. The integral of a non-zero even function over a symmetric interval (like \(-\pi\) to \(\pi\)) is generally non-zero. Thus, for a general even function, we expect \(a_0 \neq 0\) and \(a_n \neq 0\). Statements A and B are generally correct. Statement C is incorrect. - \(b_n\): The sine function, \(\sin(nx)\), is an odd function. The product of an even function (\(f(x)\)) and an odd function (\(\sin(nx)\)) is an odd function. The integral of any odd function over a symmetric interval is always zero. Thus, for any even function, \(b_n = 0\). Statement D is correct.
Step 3: Conclude the correct statements. The correct statements describing the Fourier coefficients for an even function are that \(a_0\) and \(a_n\) can be non-zero, while \(b_n\) must be zero. This corresponds to statements A, B, and D.