Question

Find the required Fourier series for the given function and sketch the graph of the function to which the series converges over three periods. $$ f(x)=\left\\{\begin{array}{ll}{0,} & {0

Short answer

In summary, the Fourier sine series for the given function is: $$ f(x) \approx \sum_{n=1}^{\infty} \frac{1}{2n}\left[2\cos\left(\frac{2n\pi}{3}\right) - \cos\left(\frac{n\pi}{3}\right) + 4\cos\left(\frac{4n\pi}{3}\right) - 4\cos\left(\frac{2n\pi}{3}\right)\right]\sin\left(\frac{n\pi x}{3\pi/2}\right) $$ When the graph is plotted using computer software, we should observe the series converging to the original function over three periods of length 6π.

Step-by-step solution

Define the Function

The given function has three branches: $$ f(x)=\left\\{\begin{array}{ll}{0,} & {0<x<\pi} \\\ {1,} & {\pi<x<2 \pi, \quad \text { sine series, period } 6 \pi} \\\ {2,} & {2 \pi<x<3 \pi}\end{array}\right. $$ Our goal is to find the sine series of the function with a period of 6π.

Find the Coefficients

To find the Fourier series coefficients, we need to compute the coefficients for each sine term. Thus, we have: $$ B_n = \frac{1}{3\pi}\int_0^{3\pi} f(x)\sin\left(\frac{n\pi x}{3\pi/2}\right) dx $$ Since the function is defined piecewise, the integral can be broken up into three parts corresponding to each region: $$ B_n = \frac{1}{3\pi}\left[\int_0^{\pi} 0\sin\left(\frac{n\pi x}{3\pi/2}\right) dx + \int_{\pi}^{2\pi} 1\sin\left(\frac{n\pi x}{3\pi/2}\right) dx + \int_{2\pi}^{3\pi} 2\sin\left(\frac{n\pi x}{3\pi/2}\right) dx\right] $$

Compute the Integrals

Evaluate each integral separately: $$ \int_0^{\pi} 0\sin\left(\frac{n\pi x}{3\pi/2}\right) dx = 0 $$ $$ \int_{\pi}^{2\pi} \sin\left(\frac{n\pi x}{3\pi/2}\right) dx = \left[-\frac{3\pi/2}{n\pi}\cos\left(\frac{n\pi x}{3\pi/2}\right)\right]_{\pi}^{2\pi} $$ $$ \int_{2\pi}^{3\pi} 2\sin\left(\frac{n\pi x}{3\pi/2}\right) dx = 2\left[-\frac{3\pi/2}{n\pi}\cos\left(\frac{n\pi x}{3\pi/2}\right)\right]_{2\pi}^{3\pi} $$ Now, add the results and substitute them back into the expression for \(B_n\): $$ B_n = \frac{1}{3\pi}\left[0 + \left(-\frac{3\pi/2}{n\pi}\cos\left(\frac{n\pi (2\pi)}{3\pi/2}\right) - (-\frac{3\pi/2}{n\pi}\cos\left(\frac{n\pi (\pi)}{3\pi/2}\right))\right) + 2\left(-\frac{3\pi/2}{n\pi}\cos\left(\frac{n\pi (3\pi)}{3\pi/2}\right) - (-\frac{3\pi/2}{n\pi}\cos\left(\frac{n\pi (2\pi)}{3\pi/2}\right))\right)\right] $$

Simplify the Coefficient Expression

Simplify the expression for \(B_n\): $$ B_n = \frac{1}{2n}\left[2\cos\left(\frac{2n\pi}{3}\right) - \cos\left(\frac{n\pi}{3}\right) + 4\cos\left(\frac{4n\pi}{3}\right) - 4\cos\left(\frac{2n\pi}{3}\right)\right] $$

Obtain the Fourier Sine Series

Now, insert \(B_n\) into the general formula for the Fourier sine series: $$ f(x) \approx \sum_{n=1}^{\infty} \frac{1}{2n}\left[2\cos\left(\frac{2n\pi}{3}\right) - \cos\left(\frac{n\pi}{3}\right) + 4\cos\left(\frac{4n\pi}{3}\right) - 4\cos\left(\frac{2n\pi}{3}\right)\right]\sin\left(\frac{n\pi x}{3\pi/2}\right) $$

Sketch the Graph

To plot the graph, we can use a computer software such as Desmos or MATLAB to plot the series with a few terms (e.g., 10 terms). Plot the graph over three periods of the sine series with a period of 6π. Remember that the series converges to the given function, so the graph should visually reflect the behavior of our given \(f(x)\) over multiple periods.

Key concepts

Piecewise Functions

Piecewise functions are a crucial part of mathematics, especially when dealing with real-world scenarios. They are functions defined by multiple sub-functions, each applicable to a different interval of the domain. In our original exercise, the function was piecewise because it took different values in distinct intervals within its period:

• From 0 to \( \pi \), the function value is 0.
• From \( \pi \) to \( 2\pi \), the function value is 1.
• From \( 2\pi \) to \( 3\pi \), the function value is 2.

The beauty of piecewise functions lies in their ability to describe complex situations through relatively simple mathematical expressions. When working with them, it’s important to recognize the endpoints of each interval and understand what value the function should take within each part.
Integrating piecewise functions can often be more complex, as each interval has to be treated separately. This means you might have to evaluate several definite integrals and combine their results, which was exactly the method employed to derive the Fourier coefficients in the solution.

Sine Series

The sine series is a specific type of Fourier series that only includes sine terms. It's particularly useful for functions that are odd about the vertical axis. Unlike general Fourier series, which involve both sine and cosine terms, a sine series captures the wave-like nature of functions
A sine series can be expressed as:

• \( f(x) = \sum_{n=1}^{\infty} B_n \sin(\frac{n\pi x}{L}) \)

The use of sine functions means you are looking at how the function's shape can be approximated by varying amplitudes and frequencies of sine waves. In the solution, the series involved finding suitable coefficients for these sine terms to properly approximate the behavior of the given piecewise function over its defined intervals.
An important aspect of the sine series in this problem was setting the appropriate period for the series. With a period of \(6\pi\), the series will accurately repeat every \(6\pi\) units, capturing the nature of the original function over several repetitions.

Periodic Functions

Periodic functions are those that repeat their values at regular intervals. The most common examples are trigonometric functions like sine and cosine. When analyzing a piecewise function with the intention of applying Fourier series, it's important to ensure the function is periodic.
Our exercise illustrates a periodic function with a specified period of \(6\pi\). This means that every \(6\pi\) units along the x-axis, the function repeats its form. In practice, this allows for the expansion in a Fourier sine series to entirely represent the behavior over this repeating interval.
Periodic functions are invaluable in various fields such as signal processing and electrical engineering. Their repeating nature allows complex waveforms to be analyzed and represented through simpler sine or cosine waves, which are the foundation of the Fourier series expansions. By using periodic functions, engineers and scientists can predict and analyze cyclic behaviors found in many natural phenomena.

Fourier Coefficients

Fourier coefficients are key to constructing Fourier series. They determine the weight or influence of each sine or cosine term in the series representation of a function. In the context of a sine series, as in our exercise, we compute only the Fourier sine coefficients \(B_n\).
The formula to find these coefficients is:
\[ B_n = \frac{1}{L} \int_0^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \]
In particular, these coefficients carry the information on how much of a particular sine wave frequency is present in the function's periodic behavior. Calculating \(B_n\) often involves integrating the product of the function and a sine function over one period.
In the step-by-step solution, the function was broken down according to its piecewise intervals to compute the integral part necessary for each \(B_n\). After deriving these coefficients and simplifying the expressions, they were used to construct the sine series that approximates the original piecewise function over its designated period.