Question

For each real \(p,-\pi \leq p \leq \pi\), find the Fourier series of the function $$ f_{p}(x)= \begin{cases}0, & -\pi \leq x \leq p, \\ 1, & p

Short answer

The Fourier series is \(\frac{\pi - p}{2\pi} + \sum_{n=1}^{\infty} \left( \frac{\sin(np)}{n\pi} \cos(nx) + \frac{(-1)^n - \cos(np)}{n\pi} \sin(nx) \right)\).

Step-by-step solution

Determine the Interval Periodicity

The function \( f_p(x) \) is defined on the interval \([-\pi, \pi]\), which implies the function has a period \( T = 2\pi \). The Fourier series needs to be calculated based on this periodic interval.

Fourier Series Expression

A function with period \(2\pi\) can be expressed as a Fourier series: \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \] where \(a_n\) and \(b_n\) are the Fourier coefficients.

Calculate \(a_0\)

The constant term \(a_0\) is determined by integrating one period of the function: \[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f_p(x) \, dx = \frac{1}{2\pi} \int_{p}^{\pi} 1 \, dx = \frac{1}{2\pi} (\pi - p) = \frac{\pi - p}{2\pi} \]

Calculate \(a_n\)

The coefficient \(a_n\) is calculated using:\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f_p(x) \cos(nx) \, dx = \frac{1}{\pi} \int_{p}^{\pi} \cos(nx) \, dx \]Evaluating this integral, we find:\[ a_n = \frac{1}{n\pi} \left( \sin(np) - \sin(n\pi) \right) = \frac{1}{n\pi} \sin(np) \](since \(\sin(n\pi) = 0\)).

Calculate \(b_n\)

The coefficient \(b_n\) is obtained by:\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f_p(x) \sin(nx) \, dx = \frac{1}{\pi} \int_{p}^{\pi} \sin(nx) \, dx \] Evaluating this integral gives:\[ b_n = \frac{1}{n\pi} \left( -\cos(np) + \cos(n\pi) \right) = \frac{1}{n\pi} \left( -\cos(np) + (-1)^n \right) \]

Assemble the Fourier Series

Combining the results, the Fourier series of \( f_p(x) \) is:\[\frac{\pi - p}{2\pi} + \sum_{n=1}^{\infty} \left( \frac{\sin(np)}{n\pi} \cos(nx) + \frac{(-1)^n - \cos(np)}{n\pi} \sin(nx) \right)\]

Key concepts

Periodicity

In mathematics, periodicity refers to the property of a function to repeat its values at regular intervals, forming cycles. A function is periodic if there exists a positive value, known as the period, such that the function's value repeats after this interval. For example, consider the function provided in the exercise, which is defined on the interval \([-\pi, \pi]\). Here, the function repeats after the interval is completed, hence it has a period of \(T = 2\pi\).
The concept of periodicity is fundamental in the analysis of Fourier series because these series are used to break down periodic functions into their basic components - sine and cosine waves, themselves inherently periodic. Recognizing the period of a function helps determine how these series will capture the behavior of the function over its effective range. Understanding periodicity is crucial when trying to represent a function with a Fourier series, allowing us to determine the correct interval over which to expand the series.

• Periodicity helps identify how frequently a function's values repeat.
• Knowing the period allows for accurate representation in Fourier series.
• On identifying the period as \(2\pi\), we can proceed with Fourier analysis.

Fourier Coefficients

Fourier coefficients, specifically \(a_n\) and \(b_n\), play a key role in forming the Fourier series representation of a periodic function. These coefficients determine the weight or contribution of each sine and cosine component in building the function back from its periodic base waves.
There are three main types of Fourier coefficients:

• Constant term \(a_0\): It's calculated as the average value of the function over a period. It helps determine the series' vertical position or baseline.
• Cosine coefficients \(a_n\): Calculated as integral projections of the function onto cosines. They represent how much of the cosine wave is needed to fit the function's pattern.
• Sine coefficients \(b_n\): Similarly, these coefficients are integral projections onto sine waves, capturing how the function shows sine-based behaviors.

For the given function \(f_p(x)\), these coefficients become:

• \(a_0 = \frac{\pi - p}{2\pi}\): This is the average or mean value of the function when evaluated over one period.
• \(a_n = \frac{1}{n\pi} \sin(np)\): This shows how each cosine component fits into the function.
• \(b_n = \frac{1}{n\pi} \left( (-1)^n - \cos(np) \right)\): This describes how sine components contribute to modeling the function.

Integral Calculus

Integral calculus is a vital mathematical tool used to compute quantities like areas, volumes, and sums, and in the context of Fourier series, it helps us determine the Fourier coefficients. Integrals allow us to calculate the continuous sum of values, essentially used here to project functions onto sine and cosine basis functions.
In the exercise, integrals are employed to determine each coefficient:

• The integral for \(a_0\) calculates the mean of the function over a period. This gives an understanding of the overall level or offset of the series.
• The calculation of \(a_n\) and \(b_n\) involves integrals of the given function multiplied by \(\cos(nx)\) and \(\sin(nx)\) respectively. These operations are a way of determining how closely the function correlates with each of these sinusoids.

This process requires knowledge of definite integrals as it evaluates the exact contributions of sine and cosine functions over the interval \([-\pi, \pi]\). Integral calculus enables us to construct an accurate and complete Fourier series by evaluating these key components.

• Integrals provide the foundation for calculating average values over intervals.
• They help decompose complex functions into simpler waveform components.
• Understanding integrals is essential for students to excel in Fourier analysis.