The question is asking for the correct formula for Parseval's relation in the Fourier series expansion. Let's break this down and analyze the options based on Parseval's theorem.

Parseval's Theorem in Fourier Series

For a function $f(x)$ defined over the interval $[-L, L]$, its Fourier series can be written as: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right)\right)$

Parseval's theorem states that the total energy (or squared norm) of the function over the interval $[-L, L]$ is equal to the sum of the squares of its Fourier coefficients. Mathematically, this is expressed as: $\int_{-L}^{L} (f(x))^2 dx = L \left( \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \right)$

Analyzing the Options

Looking at the given options:

• Option (a): $\int_{-L}^{L} (f(x))^2 dx = L \left( \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \right)$ This is exactly the correct form of Parseval's relation. This matches the equation of Parseval's theorem.

• Option (b): $\int_{-L}^{L} (f(x))^2 dx = L \left( \frac{a_0^2}{2} + \sum_{n=1}^{\infty} a_n^2 \right)$ This is incorrect because it is missing the $b_n^2$ terms in the summation.

• Option (c): $\int_{-L}^{L} (f(x))^2 dx = \frac{L}{2} \left( a_0^2 + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \right)$ This form is incorrect because the $L$ should not be divided by 2. Parseval's theorem states that it should be $L \times \left( \frac{a_0^2}{2} + \cdots \right)$.

• Option (d): $\int_{L}^{L} (f(x))^2 dx = \left[ \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \right]$ This is incorrect because the integral limits are wrong. The integral should be from $-L$ to $L$, and also it should be multiplied by $L$ outside the brackets.

Correct Answer:

The correct option is (a).

Let me know if you would like further details, or if you have any questions.

Here are 5 related questions:

1. What is the significance of Parseval's theorem in signal processing?
2. How does Parseval's theorem relate to the concept of energy in a signal?
3. Can Parseval's theorem be applied to non-periodic functions?
4. How are the Fourier coefficients $a_n$ and $b_n$ calculated for a given function?
5. What are some real-world applications of Fourier series in engineering?

Tip: Parseval's theorem is a key result linking time-domain and frequency-domain representations of signals, showing that the total energy remains conserved in both domains.