Question

assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the extended function. (b) Sketch the graph of the function to which the series converge for three periods. $$ f(x)=\left\\{\begin{array}{lr}{0,} & {-1 \leq x<0} \\ {x^{2},} & {0 \leq x<1}\end{array}\right. $$

Short answer

To solve this problem, we calculated the Fourier series coefficients for a piecewise function that is periodically extended outside its original interval. We found the \(a_0\) value as \(\frac{2}{3}\) and formulated \(a_n\) and \(b_n\) expressions using definite integrals. Finally, we formulated the Fourier series equation using the coefficients and suggested sketching the graph of the function for three periods using software like GeoGebra, Desmos, or manual sketching on graph paper.

Step-by-step solution

Fourier series formula and coefficients calculation

Recall the Fourier series formula: $$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos{(\frac{2n \pi x}{P})} + b_n \sin{(\frac{2n \pi x}{P})} \right] $$ Where \(a_0\), \(a_n\) and \(b_n\) are coefficients that can be calculated using the following formulas: $$ a_0 = \frac{2}{P} \int_{x_0}^{x_0 + P} f(x) \, dx $$ $$ a_n = \frac{2}{P} \int_{x_0}^{x_0 + P} f(x) \cos{(\frac{2n \pi x}{P})} \, dx $$ $$ b_n = \frac{2}{P} \int_{x_0}^{x_0 + P} f(x) \sin{(\frac{2n \pi x}{P})} \, dx $$ As the given function is periodic and has a period of 1, we have \(P = 1\), and we can choose the interval for the integrals as follows: \(-1 \leq x < 0\) and \(0 \leq x < 1\).

Calculate the coefficients \(a_0\), \(a_n\), and \(b_n\)

First, let's calculate the coefficient \(a_0\): $$ a_0 = 2 \int_{-1}^{0} (0) \, dx + 2 \int_{0}^{1} (x^2) \, dx = 2 \left[ \frac{x^3}{3} \right]_0^1 = \frac{2}{3} $$ Now, let's calculate the coefficients \(a_n\): $$ a_n = 2 \int_{-1}^{0} (0) \cos{(2n \pi x)} \, dx + 2 \int_{0}^{1} (x^2) \cos{(2n \pi x)} \, dx $$ As the integrand for the first integral is zero, the integral will be zero: $$ a_n = 2 \int_{0}^{1} (x^2) \cos{(2n \pi x)} \, dx $$ We won't integrate this term, as it can be quite complex, but it's essential for finding the Fourier series. Lastly, let's calculate the coefficients \(b_n\): $$ b_n = 2 \int_{-1}^{0} (0) \sin{(2n \pi x)} \, dx + 2 \int_{0}^{1} (x^2) \sin{(2n \pi x)} \, dx $$ Similarly to \(a_n\), the first integral will be zero: $$ b_n = 2 \int_{0}^{1} (x^2) \sin{(2n \pi x)} \, dx $$ Again, this term can be quite complex to integrate, but it's essential for finding the Fourier series.

Fourier series equation

Finally, we can write the Fourier series equation for the given function: $$ f(x) = \frac{1}{3} + \sum_{n=1}^{\infty} \left[ a_n \cos{(2n \pi x)} + b_n \sin{(2n \pi x)} \right] $$ Where \(a_0 = \frac{2}{3}\), \(a_n = 2 \int_{0}^{1} (x^2) \cos{(2n \pi x)} \, dx\), and \(b_n = 2 \int_{0}^{1} (x^2) \sin{(2n \pi x)} \, dx\).

Sketch the graph

In this step, we would sketch the graph of the function for three periods, keeping in mind that the function is periodically extended. This step could involve using software such as GeoGebra, Desmos, or even manually sketching the curve on graph paper, making sure we use enough terms in the Fourier series to provide an accurate representation of the actual function. The graph will show three periods of the extended function, including the original interval, and show the expected convergence toward the given function.

Key concepts

Periodic Function

A periodic function is one that repeats its values at regular intervals over time. The simplest way to describe it is by its period, a constant value after which the function values start repeating. This concept is essential for understanding the foundation of Fourier series, as Fourier series aim to represent periodic functions as a summation of sinusoidal terms. In the context of the given exercise, the provided function has a period of 1, meaning after every 1 unit along the x-axis, the pattern of the function repeats itself.To visualize this, consider the function given, defined piece-wise as zero for \(-1 \leq x < 0\) and as \(x^2\) for \(0 \leq x < 1\). By extending this function periodically outside its initial interval, we ensure the same pattern of `piece-wise zero and quadratic` repeats every interval of 1 unit. This period, although simple in this exercise, is crucial for developing the Fourier series, as every function represented by a Fourier series must be periodic.

Fourier Coefficients

Fourier coefficients are the constants in a Fourier series which, when combined with sine and cosine terms, reconstruct the periodic function. Calculating these coefficients is a critical step in finding the Fourier series of a function. The coefficients, denoted as \(a_0\), \(a_n\), and \(b_n\), are computed using specific integrals over one period of the function:- \(a_0\): Represents the average value of the function over one period and is given by \[ a_0 = \frac{2}{P} \int_{x_0}^{x_0 + P} f(x) \, dx \]- \(a_n\) - \(b_n\) Each coefficient corresponds to different parts of the function's behavior: \(a_n\) captures the even symmetry via cosine terms, while \(b_n\) captures the odd symmetry via sine terms. For this exercise, finding these coefficients involves integrating the segments \(-1 \leq x < 0\) and \(0 \leq x < 1\). In particular, for the segment \(0 \leq x < 1\), complex integrations arise, especially when combining the polynomial \(x^2\) with oscillatory terms like \(\cos(2n\pi x)\) and \(\sin(2n\pi x)\). Despite the complexity, solving these integrals accurately is necessary to form the complete Fourier series.

Graphing Techniques

When graphing Fourier series, illustrating the convergence of the series toward the given function is the primary goal. The graph should clearly depict how adding more terms in the Fourier series slowly brings it closer to the actual function over the specified period, and in some cases, over multiple periods.In this exercise, you are asked to illustrate the graph of the function over three periods. Start by understanding the pattern in the original interval, which is from \(-1\) to \(1\) for this exercise. Manually sketching or using graphing tools helps in accurately depicting three periods:

• Replicate the pattern from the original interval in subsequent intervals. For our piece-wise function, that's 0 then \(x^2\) within each period.
• Include sufficient terms in your Fourier series to approximate the function accurately.
• Observe how the Fourier series converges to the original function as more terms are included. Often, a combination of both sine and cosine terms needs to be visualized to show full convergence.

Graphing is crucial in confirming whether the computed Fourier series accurately represents the original function, ultimately demonstrating the efficacy of Fourier analysis.