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. $$ f(x)=\left\\{\begin{array}{ll}{x,} & {0 \leq x<1} \\ {1,} & {1 \leq x<2}\end{array} \quad \text { sine series, period } 4\right. $$

Short answer

Question: Find the Fourier sine series of the given function over a period of 4 and sketch its graph for three periods. The function is f(x) = x for 0 ≤ x < 1, and f(x) = 1 for 1 ≤ x < 2. Solution: The Fourier sine series of the given function is $$ f(x) \sim \sum_{n=1}^{\infty} \left(-\frac{4}{n^2\pi^2}(1 - (-1)^{n}) + \frac{2}{n\pi}(1 - \cos{n\pi})\right) \sin{\frac{n\pi x}{2}} $$ To sketch the graph of the Fourier sine series over three periods, consider a few of the first terms (e.g., 5-10 terms) and plot f(x) for values of x in the range of -4 to 8 using a graphing software. The resulting graph will approximate the original function and will be an odd function with respect to the y-axis.

Step-by-step solution

Extend the function for a period of 4

We need to extend the given function f(x) for a period of 4, with f(x) being equal to zero outside [0, 2]. Therefore, f(x) becomes as follows: $$ f(x)=\left\\{\begin{array}{ll}{x,} & {0 \leq x<1} \\\ {1,} & {1 \leq x<2}\end{array}\right. $$ And, $$ f(x)=0\textrm{ , } \textrm{elsewhere} $$

Fourier Sine Series

Since we only need the Fourier sine series, we will only use the sine terms of the general Fourier series formula. The Fourier sine series representation of a function is given as: $$ f(x) \sim \sum_{n=1}^{\infty} B_n \sin{\frac{n\pi x}{L}} $$ Where, $$ B_n = \frac{2}{L} \int_{0}^{L} f(x) \sin{\frac{n\pi x}{L}} dx $$ In our case, f(x) has a sine series with period 4, so L = 2. So, $$ B_n = \int_{0}^{2} f(x) \sin{\frac{n\pi x}{2}} dx $$

Calculate the Bn coefficients

Now, we'll calculate the B_n coefficients for the given function. Since f(x) has two different parts, we need to break the integration into two parts: First part (0 ≤ x < 1): $$ B_n^{(1)} = \int_{0}^{1} x \sin{\frac{n\pi x}{2}} dx $$ Second part (1 ≤ x < 2): $$ B_n^{(2)} = \int_{1}^{2} \sin{\frac{n\pi x}{2}} dx $$ Using integration by parts for the first part, $$ B_n^{(1)} = -\frac{4}{n^2\pi^2}(1 - (-1)^{n}) $$ For the second part, $$ B_n^{(2)} = \frac{2}{n\pi}(1 - \cos{n\pi}) $$ Adding them together, $$ B_n = B_n^{(1)} + B_n^{(2)} = -\frac{4}{n^2\pi^2}(1 - (-1)^{n}) + \frac{2}{n\pi}(1 - \cos{n\pi}) $$ Finally, the Fourier sine series for f(x) becomes: $$ f(x) \sim \sum_{n=1}^{\infty} \left(-\frac{4}{n^2\pi^2}(1 - (-1)^{n}) + \frac{2}{n\pi}(1 - \cos{n\pi})\right) \sin{\frac{n\pi x}{2}} $$

Graph the Fourier sine series

In order to sketch the graph of the Fourier sine series over three periods, we'll truncate the series by considering a few of the first terms (5-10 terms), and plot f(x) for values of x in the range of -4 to 8. You can plot the graph using a software like Wolfram Alpha, GeoGebra, or Desmos. The resulting graph will approximate our original function, and because we have used a sine series, it will be an odd function with respect to the y-axis.

Key concepts

Fourier Series Convergence

Understanding the concept of Fourier series convergence is crucial when dealing with Fourier sine series. A Fourier series is a way to represent a function as an infinite sum of sine and cosine functions. This series converges to the original function under certain conditions. Particularly, the Fourier series converges to the function at points where it is continuous. If there's a discontinuity, the series converges to the midpoint of the jump. In the given exercise, the function is piecewise continuous and well-defined within the interval, making convergence an applicable concept. One might find it useful to visualize the series as a sum of waves of different frequencies that, when added together, sculpt out the shape of the original function.

For educational purposes, recognizing where and how the series converges can significantly aid in understanding the behavior of the function and its representation. This will be important when we sketch the graph, as visualizing the approximated behavior over several periods can provide insight into the nature of the function's convergence.

Fourier Coefficients Calculation

Fourier coefficients calculation is a vital step in obtaining the Fourier sine series of a function. These coefficients determine the amplitude of each sine (or cosine) wave in the series. The calculation involves integration, and as seen in the exercise, it's performed over one period of the function. In this case, for a sine series with a period of 4, we calculate the coefficients using the given formula for \( B_n \).

As per the exercise solution, you can see the Fourier coefficients being calculated in two parts because the function has two distinct behaviors in different intervals. This segmentation and subsequent integration reflect the piecewise nature of the function and ensure that each part contributes correctly to the Fourier series. To simplify the calculation, students can utilize methods such as integration by parts, which is demonstrated in the subsequent steps of the exercise.

Integration by Parts

The technique of Integration by Parts is derived from the product rule of differentiation and provides a method to integrate products of functions. It expresses the integral of a product of two functions as the product of the functions minus the integral of their product's derivative. In our exercise, integration by parts is used to simplify the coefficient \( B_n^{(1)} \) for the first part of the function.

Students learning Fourier series will benefit from mastering integration by parts as it often simplifies complex integrals. When applying integration by parts to the given exercise, the student can discern how the method effectively breaks down the product of \( x \text{ and }\frac{n\text{π}}{2} \sin(\frac{n\text{π}x}{2}) \), facilitating the calculation of \( B_n^{(1)} \). Including step-by-step application of this method can greatly aid in understanding and can be a profound tool in the student's mathematical toolkit.

Mathematical Graph Plotting

The final step in the exercise is Mathematical Graph Plotting, which is a visual representation of the function and its Fourier sine series. Graphing is an essential part of understanding the behavior and convergence of the series, especially over multiple periods. By plotting the first few terms of the series, students can observe how the sum begins to approximate the original function.

It's important to recognize the symmetry in the sine series, where plotting beyond the original interval will showcase the odd function behavior around the y-axis. This is particularly insightful as it provides a visual affirmation of the convergence and approximation behaviors that the Fourier sine series exhibits. Tools like GeoGebra, Desmos, or graphing calculators can facilitate this process, allowing for a clear visual comparison between the graph of the original function and the Fourier series approximation. Encouraging students to graph the series themselves can provide a tangible connection to the underlying mathematical concepts.