Question

In each of Problems 15 through 22 find the required Fourier series for the given function and sketch the graph of the function to which the series converges over three periods. $$ \begin{array}{l}{f(x)=\left\\{\begin{array}{ll}{1,} & {0

Short answer

The graph of the Fourier series representation will oscillate between the values of 1 and 0 over each period. In intervals [0, 1], [4, 5], and [8, 9], the function will display oscillatory wave-like behavior as it converges to the value 1. In contrast, the function will remain close to 0 in intervals [1, 2], [5, 6], and [9, 10]. As more terms are included in the Fourier series, the graph will more closely resemble the original piecewise function.

Step-by-step solution

Calculate the Fourier series coefficients (an)

To find the coefficients an, we use the following formula: $$ a_n = \frac{2}{T} \int_0^T f(x) \cos \left(\frac{2n\pi x}{T} \right) dx $$ where T is the period of the function, which is given as 4. Since our function f(x) is piecewise, we will separate the integral into the two intervals: $$ a_n = \frac{2}{4} \left[\int_0^1 \cos \left(\frac{2n\pi x}{4} \right) dx + \int_1^2 0 \cos \left(\frac{2n\pi x}{4} \right) dx \right] $$ Solve the integral to find the coefficients an: $$ a_n = \frac{1}{2} \left[\int_0^1 \cos \left(\frac{n\pi x}{2} \right) dx + 0 \right] = \frac{1}{2} \int_0^1 \cos \left(\frac{n\pi x}{2} \right) dx $$ $$ a_n = \frac{1}{n\pi} \left[\sin \left(\frac{n\pi x}{2} \right) \right]_0^1 = \frac{1}{n\pi} (\sin(n\pi / 2) - 0) $$

Write down the Fourier cosine series representation of the function f(x)

With the coefficients an found, we can now write the Fourier series representation of f(x) as: $$ f(x) = \frac{1}{2}a_0 + \sum_{n=1}^{\infty} a_n \cos \left(\frac{2n\pi x}{T} \right) $$ Plug in the coefficients an: $$ f(x) = \frac{1}{2} \cdot \frac{1}{n\pi}(\sin(n\pi / 2) - 0) + \sum_{n=1}^{\infty} \frac{1}{n\pi} (\sin(n\pi / 2) - 0) \cos \left(\frac{n\pi x}{2} \right) $$ Simplify the expression: $$ f(x) = \sum_{n=1}^{\infty} \frac{1}{n\pi} \sin\left(\frac{n\pi}{2} \right) \cos \left(\frac{n\pi x}{2} \right) $$

Sketch the graph of the function to which the series converges over three periods

With the Fourier series representation of f(x) found, we can now sketch the graph of the function. Since the given function is piecewise, we will have alternating intervals of 1 and 0. Because of the cosine terms in our Fourier series, the graph will converge to an oscillating wave-like function that seeks to approximate the piecewise function f(x). Over each period, the oscillation will go from a peak value to a minimum value and back again, ultimately converging to the piecewise function f(x) as the number of terms included in the series increases. Sketching the graph of our function for three periods, we will see alternating "wave-like" behavior in the intervals of [0, 1], [4, 5], and [8, 9] when the function is equal to 1, and a zero-valued function in [1, 2], [5, 6], and [9, 10] intervals when the function is equal to 0.

Key concepts

Cosine Series

A Fourier series can break down a complex, periodic function into a simpler form made up of sines and cosines. Cosine series refer specifically to decompositions that utilize only cosine terms. This is particularly useful when dealing with even functions, though they can be applied to any piecewise function as well.

In the case we're looking at, the piecewise function is represented as a cosine series. The reason for using only cosine terms is often linked to the nature of the interesting properties of cosines:

• Symmetry: Cosine functions are symmetrical about the y-axis.
• Efficiency: Cosine series are highly efficient in capturing the behavior of even or symmetric parts of signals.

Understanding the cosine series helps us to simplify periodic functions and represent them using a sum of cosine terms, making analysis and transformations far easier. In practice, you will often first find the required cosine coefficients (here termed as \(a_n\)), which are derived by integrating over a single period, and then use these to express the entire function as a series.

Piecewise Function

A piecewise function is a function composed of multiple sub-functions, each applying to a certain interval in its domain. There might be different formulas for different parts of the function's input range. In this exercise, we encounter a piecewise function defined over two distinct sub-intervals:

• For \(0 < x < 1\), the function is 1.
• For \(1 < x < 2\), the function is 0.

The definition of a piecewise function is crucial since it allows for a more detailed and accurate description of functions that have distinct expressions over their domains. This type of function is especially useful in Fourier analysis, helping us understand how complex signals can vary across different intervals. For instance, by separating function evaluations into manageable sections, piecewise functions can effectively model real-world phenomena like signal transmissions or mechanical vibrations.

Periodicity

The concept of periodicity refers to the repeating nature of wave-like functions over specified intervals known as periods. Periodicity plays a significant role in Fourier series analysis because it allows for any periodic function to be expressed as a sum of sines and cosines whose periods are harmonics of the function's period.

In the exercise, we're given a period of 4. This means that every segment of the function returns to its starting point after 4 units along the x-axis. This repetitive behavior is key in Fourier series decomposition. Through periodicity, we ensure that our cosine series representation accurately mirrors the original function, even when extended beyond a single period. Indeed, periodicity ensures the approach maintains cognitive and analytical consistency, allowing us to think about entire cycles rather than just individual functions.

Graphical Representation

Graphical representation is crucial for visualizing the outcome of Fourier series expansion. It provides a clear picture of how well the series approximates the original function, particularly over multiple periods. By sketching, you can immediately see alterations in intervals as they oscillate between values.

In this specific case, the graphical representation should show the piecewise function approximated by a series of cosine waves. The exercise indicates a graph over three periods where values alternate between active (1) and inactive (0). This makes the task of sketching visually straightforward yet informative. The behavior results in repeated rises and falls within each segment, demonstrating how closer approximations match more terms in the Fourier series. Seeing input loops in a physical form solidifies conceptual understanding and enhances analytical interpretations when engaging with functions.