91Ó°ÊÓ

A periodic function is one that repeats its values at regular intervals or periods. In the context of trigonometry and Fourier series, periodic functions are foundational. Understanding the periodic nature of a function allows us to represent it simply and predictably over any length.

In practical terms, if a function satisfies the frequency condition such that \( f(x+T) = f(x) \) for all \( x \), where \( T \) is the period, it's classified as periodic. This repetition is crucial when expanding functions like \( f(x) = 1-x^2 \) to encompass the entire real line, particularly when extended beyond its initial interval of \(-1\) to \(1\).

In such expansions, approximations using Fourier series help understand the behavior over several periods. This enables easier visualization and manipulation in various applications, such as engineering and physics.

An even function is symmetric about the y-axis. This special symmetry property simplifies many mathematical operations, especially in Fourier analysis.

An even function satisfies the property \( f(-x) = f(x) \). The function given in the exercise, \( f(x) = 1 - x^2 \), is a perfect example. By substituting \( -x \) into the function, we see that it remains unchanged. This means it exhibits even symmetry.

This symmetry is beneficial because, for even functions, only the cosine terms of a Fourier series are needed. This reduces computations significantly, as only half of the usual Fourier coefficients (those for sine terms) need to be calculated. Thus, expanding an even function into a cosine series becomes both simpler and more efficient.

A cosine series is a type of Fourier series used to express an even function. Since even functions are symmetric, they perfectly align with the symmetry of cosine, which is also even.

In practice, a cosine series is composed solely of cosine terms. It captures the essence of the original even function using the formula:

• \( f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n \pi x}{L}\right) \)

where \( L \) is half the period.

In our example, since the interval \([-1, 1]\) makes \( L = 1 \), we only have cosine terms in the expansion. This effectively mirrors and replicates the function's behavior over several periods, constructing an accurate representation through the series.

Calculating coefficients is an integral part of constructing a Fourier series. For an even function, only the \( a_0 \) and \( a_n \) coefficients need to be determined, which simplifies the computational process.

The coefficient \( a_0 \) represents the average value of the function over one period. For our function, it was calculated as follows:

• \( a_0 = \frac{2}{1} \int_{0}^{1} (1-x^2) dx = \frac{4}{3} \)

For \( a_n \), which represents the amplitudes of the cosine terms, we usually employ the integral:

• \( a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n \pi x}{L}\right) dx \)

In cases like our example, when the integral becomes complex, alternative formulas or properties, such as \( a_n = \frac{4}{n^2 \pi^2} (1 - (-1)^n) \), are used to simplify calculation. These coefficients are crucial as they quantify the contribution of each cosine term, building the function's full periodic expression.