Question

(a) Find the required Fourier series for the given function. (b) Sketch the graph of the function to which the series converges for three periods. (c) Plot one or more partial sums of the series. $$ f(x)=2-x^{2}, \quad 0

Short answer

Question: Find the Fourier sine series of the function f(x) = 2-x^2 in the interval 0 < x < 2 with period 4, and sketch the graph for three periods. Answer: The Fourier sine series for the function f(x) = 2-x^2 in the given interval and with the given period is: $$ f(x) = \sum_{n=1}^{\infty} \frac{8}{\pi^2 n^2} (-1)^{n+1} \sin \left(\frac{n \pi x}{2} \right) $$ To sketch the graph for three periods, plot the original function f(x) = 2-x^2 in the interval 0 < x < 2, then repeat the graph for the next two periods in the intervals 2 < x < 4 and 4 < x < 6. To plot one or more partial sums of the series, calculate the first few terms of the series (for example, n=1, n=2, and n=3 terms) and plot them on the same graph to get an idea of how the series converges to the original function as more terms are included.

Step-by-step solution

Find the general formula for Fourier sine series

The general formula for the Fourier sine series of a function f(x) in the interval 0 < x < L with period 2L is given by: $$ f(x) = \sum_{n=1}^{\infty} B_n \sin \left(\frac{n \pi x}{L} \right) $$ Where the Fourier coefficients B_n are given by: $$ B_n = \frac{2}{L} \int_0^L f(x)\sin\left(\frac{n\pi x}{L}\right) dx $$

Determine the Fourier coefficients for the given function

To determine the Fourier coefficients B_n for f(x) = 2-x^2 with L = 2, we need to substitute f(x) and L into the equation for B_n: $$ B_n = \frac{2}{2} \int_0^2 (2-x^2) \sin \left(\frac{n \pi x}{2} \right) dx $$ Simplify to get: $$ B_n = \int_0^2 (2-x^2) \sin \left(\frac{n \pi x}{2} \right) dx $$

Write the Fourier sine series using the calculated coefficients

Now, we need to solve the integral to find B_n, then use it to write the Fourier sine series for f(x): $$ B_n = \int_0^2 (2-x^2) \sin \left(\frac{n \pi x}{2} \right) dx $$ Solve the integral using integration by parts and simplication, we find: $$ B_n= \frac{8}{\pi^2 n^2} (-1)^{n+1} $$ Now, we can write the Fourier sine series for f(x) as: $$ f(x) = \sum_{n=1}^{\infty} \frac{8}{\pi^2 n^2} (-1)^{n+1} \sin \left(\frac{n \pi x}{2} \right) $$

Sketch the graph of the function for three periods

To sketch the graph of the function f(x) = 2-x^2 for three periods with period 4, sketch the graph in the interval 0 < x < 2, then repeat the graph for next two periods which are in intervals 2 < x < 4 and 4 < x < 6.

Plot one or more partial sums of the series

To plot one or more partial sums of the series, simply calculate the first few terms of the series and plot them. For example, you could calculate the first, second, and third partial sums (n=1, n=2, n=3 terms of the series), and plot these on the same graph to get an idea of how the series converges to the original function as more terms are included.

Key concepts

Fourier Coefficients

Understanding Fourier coefficients is fundamental when working with Fourier series. Fourier coefficients, represented typically by the symbols 'a_n', 'b_n' for cosine and sine series respectively, are key in defining how a periodic function can be broken down into a series of sines and cosines. In the case of a sine series, you will only see coefficients like 'B_n'. The formula for a Fourier sine series coefficient 'B_n' is given by:

\[\begin{equation}B_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx\end{equation}\]
These coefficients are determined through integration and reveal a lot about the original function's behavior especially regarding its symmetry. When you calculated the coefficients for the function 'f(x) = 2 - x^2' for a sine series with period 4, you found 'B_n' to be a result of an integral solved using integration by parts, helping to express 'f(x)' in terms of the sine series.

Integration by Parts

Integration by parts is an essential tool for finding the Fourier coefficients, particularly when the function and sine or cosine function product is more complex. Derived from the product rule in differentiation, integration by parts is given by the following formula:

\[\begin{equation}\int u dv = uv - \int v du\end{equation}\]
This technique essentially reduces the problem to a simpler integral that can be solved more directly. In the solution for 'B_n', using integration by parts to integrate the product of '(2 - x^2)' and a sine function was crucial. By cleverly choosing 'u' to be '(2 - x^2)' and 'dv' to be the sine function, we simplified the calculation. This method emphasizes the power of integration by parts in solving Fourier coefficients integrals, making the whole process manageable even for complex terms.

Convergence of Fourier Series

A fundamental question in Fourier analysis is under what conditions a Fourier series converges to the original function. In general, for reasonable functions (those that are piecewise continuous), the Fourier series converges to the function at points where it is continuous. At discontinuities, it converges to the midpoint of the discontinuity's jump. It is remarkable when considering periodic phenomena or signal processing, the series can approximate complex functions to any desired degree of accuracy as more terms are included.

For 'f(x) = 2 - x^2', we expect the series to converge very well since the function is continuous and even differentiable. The convergence assures us that by calculating the sine series with sufficient terms, we can get as close as we want to the original function, which is reassuring for both mathematical and practical applications.

Partial Sums of Fourier Series

Partial sums of Fourier series are practical representations of the series by summing a finite number of terms. Each partial sum is a truncated version of the full series, giving an approximation of the original function. As you increase the number of terms, the approximation improves. In the context of our function 'f(x) = 2 - x^2', plotting partial sums is handy for visualizing how the sum approaches the actual function graphically.

A partial sum up to 'n' terms is represented as:\[\begin{equation}S_n(x) = \sum_{k=1}^{n} B_k \sin \left(\frac{k \pi x}{L} \right)\end{equation}\]
By plotting 'S_1(x)', 'S_2(x)', 'S_3(x)', and so on, one can observe the convergence behavior of the Fourier series graphically. Each partial sum adds more detail to the approximation, capturing more characteristics of 'f(x)'. For students, seeing these plots can greatly aid in understanding convergence and the power of Fourier series in representing periodic functions.