Question

A periodic function f(x) = x² for -π < x < π is expanded in a Fourier series. Which of the following statement(s) is/are correct ?

• A. Coefficients of all the sine terms are zero
• B. The first term in the series is \(\frac{\pi^2}{3}\)
• C. The second term in the series is −4cosx
• D. Coefficients of all the cosine terms are zero

Answer 1

The Correct Option is A, B, C

To determine which statements are correct regarding the Fourier series expansion of the periodic function \( f(x) = x^2 \) in the interval \(-\pi < x < \pi\), let's go through the process of Fourier series expansion step-by-step.

Step 1: Fourier Series Expansion

A Fourier series for a function \( f(x) \) in the interval \(-L\) to \(L\) is given by:

\(f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)]\)

Where:

• \(a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \, dx\)
• \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) \, dx\)
• \(b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \sin(nx) \, dx\)

Step 2: Calculation of Coefficients

First, let's determine the \(a_0\) term:

\(a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \, dx = \frac{1}{\pi} \left[\frac{x^3}{3}\right]_{-\pi}^{\pi} = \frac{2\pi^3}{3\pi} = \frac{2\pi^2}{3}\) 
 

Thus, the first term is:

\(\frac{a_0}{2} = \frac{\pi^2}{3}\)

Next, let's calculate \(b_n\):

\(b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \sin(nx) \, dx = 0\)

This integral evaluates to zero because the integrand \(x^2 \sin(nx)\) is an odd function over a symmetric interval about the origin.

Now, let's determine \(a_n\):

The integral for \(a_n\) generally involves integration by parts and yields:

\(a_1 = -4\)

Step 3: Forming the Series

The Fourier series expansion becomes:

\(f(x) = \frac{\pi^2}{3} - 4\cos(x) + \text{ other cosine terms...}\)

Conclusion

Given these calculations:

• Coefficients of all the sine terms are zero: Correct, as explained \(b_n = 0\).
• The first term in the series is \(\frac{\pi^2}{3}\): Correct, as shown above.
• The second term in the series is -4cosx: Correct as \(a_1\) was calculated to be -4.
• Coefficients of all the cosine terms are zero: Incorrect, since the cosine terms do exist and are significant in the expansion.