91Ó°ÊÓ

When exploring the Fourier series representation of a function, understanding Fourier coefficients is crucial. These coefficients essentially break down the function into different frequency components, allowing us to express the function as a sum of simple oscillating functions.
In the exercise, the function given is \(f(x) = x^2\) and we need to express this using a Fourier series.
The Fourier series of a function within an interval \([-\pi, \pi]\) is generally written as:

• \(a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos(nx) + b_n \sin(nx) \right]\)

For our specific problem, which involves only cosine terms, the relevant coefficients are \(a_0\) and \(a_n\).
The constant term, \(a_0\), in this series captures the average value of the function over the interval, while \(a_n\) coefficients quantify the amplitude of each cosine term corresponding to their frequency.

The presence of cosine terms in a Fourier series indicates the periodic nature of the function, akin to how sound waves have characteristic frequencies.
In this exercise, we were tasked with finding a series representation of \(f(x) = x^2\) that involved only cosine terms.
Here's why we're focused specifically on cosines:

• Cosine functions are even functions, which mirror the symmetry properties of the function \(x^2\) over the interval \([-\pi, \pi]\).
• Because of this symmetry, the sine terms in the Fourier series vanish (i.e., \(b_n = 0\)), leaving only the cosine terms.

Thus, our series takes the form \(a_0 + \sum_{n=1}^{\infty} a_n \cos(nx)\), where each term \(\cos(nx)\) adds its unique oscillatory component to approximate \(x^2\) across the interval.

Integration by parts is an essential technique for finding Fourier coefficients, especially when the integral involves a product of functions.
In our context, this technique was crucial in computing \(a_n\) coefficients for a function \(f(x) = x^2\).
Here's the general setup:

• When integrating \(x^2 \cos(nx)\) over \([-\pi, \pi]\), we use the integration by parts formula: \( \int u \, dv = uv - \int v \, du \).
• In our case, we let \(u = x^2\) and \(dv = \cos(nx) \, dx\).
• This yields a solution for \(a_n\) as \((-1)^{n} \frac{4}{n^2}\), which precisely matches the Fourier series representation.

Integrating by parts not only provides a useful tool for solving these integrals but also unveils the relationship between the components of the series, ensuring an accurate reconstruction of the original function.