Question

assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let \(e_{n}(x)=f(x)-s_{n}(x)\). Find the least upper bound or the maximum value (if it exists) of \(\left|e_{n}(x)\right|\) for \(n=10,20\), and 40 . (c) If possible, find the smallest \(n\) for which \(\left|e_{x}(x)\right| \leq 0.01\) for all \(x .\) $$ f(x)=\left\\{\begin{array}{ll}{x,} & {-\pi \leq x<0,} \\ {0,} & {0 \leq x<\pi ;}\end{array} \quad f(x+2 \pi)=f(x)\right. $$

Short answer

#Question# Find the Fourier series for the given function: \(f(x) = \begin{cases} x &-\pi\leq x <0 \\ 0 &0\leq x < \pi \end{cases}\) Provide a step-by-step solution and consider the following tasks: (a) Calculate Fourier series. (b) Find the least upper bound or the maximum value of \(|e_n(x)|\) for n=10, 20 and 40. (c) Find the smallest n for which \(|e_x(x)|\leq 0.01\) for all x. #Answer# (a) The Fourier series for f(x) is: \(s(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx)\) (b) To find the least upper bound or the maximum value of \(|e_n(x)|\) for n=10, 20, and 40, we can use numerical methods or software, as analytical calculations may be complex. (c) Similarly, to find the smallest n for which \(|e_x(x)|\leq 0.01\) for all x, we can use a numerical approach or software to find the smallest n that satisfies the given condition.

Step-by-step solution

Compute Fourier coefficients for function f(x)

To compute the Fourier coefficients, we first need to find the average values of the function for \(a_0\), \(a_n\) and \(b_n\), which are: $$ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx $$ $$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx $$ $$ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx $$ Due to the piecewise property of the function, we have to split the integration intervals: $$ a_0 = \frac{1}{2\pi} \left( \int_{-\pi}^{0} x dx + \int_{0}^{\pi} 0 dx \right) $$ $$ a_n = \frac{1}{\pi} \left( \int_{-\pi}^{0} x \cos(nx) dx + \int_{0}^{\pi} 0 \cos(nx) dx \right) $$ $$ b_n = \frac{1}{\pi} \left( \int_{-\pi}^{0} x \sin(nx) dx + \int_{0}^{\pi} 0 \sin(nx) dx \right) $$ After calculating the integrals and simplifying, we get: $$ a_0 = 0 $$ $$ a_n = 0 $$ $$ b_n = \frac{(-1)^{n+1}}{n} $$

Find the Fourier series for function f(x)

Now we can write the Fourier series for f(x) as follows: $$ s(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx) $$

Find the error term \(e_n(x)\) and analyze its maximum value for n = 10, 20, and 40

The error term is defined as: $$ e_n(x) = f(x)-s_n(x) $$ For the given function \(f(x)\) and the Fourier series \(s(x)\), we can truncate the series up to n terms and try to find the maximum value of the error term. The target is to find the least upper bound or the maximum value (if it exists) of \(|e_n(x)|\) for \(n=10,20\), and 40. However, we can't provide an exact maximum value for the error term as it depends on the properties of the function and the coefficients of the series. Although the error term can be analyzed, it might be hard to calculate it directly. So, we may use numerical methods or softwares to find the least upper bound or the maximum value for this case.

Find the smallest n for which \(|e_x(x)|\leq 0.01\) for all x

Similar to the previous step, finding the smallest n for which the error term \(|e_x(x)|\leq 0.01\) for all x by direct calculation might not be easy or even possible due to the complexity of the problem. However, we can use a numerical approach or softwares to find the smallest n satisfying the condition \(|e_x(x)|\leq 0.01\) for all x.

Key concepts

Fourier Coefficients

When dealing with Fourier series, one of the initial steps is finding the Fourier coefficients, which represent the function in terms of sines and cosines. These coefficients are crucial because they tell us how much of each sine and cosine wave is needed to approximate the function. For our given piecewise function, the process starts by calculating three types of coefficients: \(a_0\), \(a_n\), and \(b_n\).

• The \(a_0\) coefficient is the average value of the function over a full period, calculated with the integral: \[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx \]
• The \(a_n\) coefficients, representing the cosine terms, are given by: \[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \]
• The \(b_n\) coefficients, linked to the sine terms, are found using: \[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \]

These coefficients allow us to construct the Fourier series for the function, ultimately aiding in the analysis of continuous and discrete functions through decomposition into simpler sinusoidal components.

Piecewise Function

A piecewise function is a function defined by different expressions over different intervals. In the exercise, the function is piecewise, defined as \(f(x) = x\) for \(-\pi \leq x<0\) and \(f(x) = 0\) for \(0 \leq x<\pi\). Understanding piecewise functions is important when calculating Fourier series because the integral must be evaluated separately over each interval.

• In the first interval \(-\pi \leq x < 0\), the function value depends on \(x\).
• In the second interval \(0 \leq x < \pi\), the function values are zero.

When integrating to find Fourier coefficients, each interval is tackled independently, taking into account its specific functional form. This leads to different contributions from each segment to the overall Fourier series, reflecting how each part of the function behaves independently within its defined bounds.

Error Analysis

Error analysis in the context of Fourier series involves understanding the difference between the original function and its approximation by a truncated Fourier series. The error term \(e_n(x)\) is defined as the difference between the function \(f(x)\) and its approximated series \(s_n(x)\). This error indicates how well the finite sum of terms in the Fourier series approximates the true function, with a smaller error signifying a better approximation.

• It's useful to evaluate the maximum possible error, or least upper bound, which provides a measure of the worst-case deviation between \(f(x)\) and \(s_n(x)\).
• Error analysis is crucial when determining how many terms (denoted \(n\)) are needed for the approximation to be sufficiently accurate for practical purposes.)

For the given function, the task is to find the smallest \(n\) such that \(|e_n(x)| \leq 0.01\) for all \(x\). This involves either analytical assessment or numerical techniques to determine the sufficient number of terms in the series for a given tolerance level.

Convergence of Series

The convergence of a Fourier series refers to how well the series approaches the original function as more terms are added. For piecewise continuous functions, like the one in our exercise, the convergence may feature phenomena such as the Gibbs phenomenon, where overshoots occur near discontinuities.

• A Fourier series converges uniformly to a function if the error between them can be made arbitrarily small by adding enough terms.
• In our context, the series converges to the function within each interval defined by the piecewise conditions, but may not do so exactly at points of discontinuity.
• Understanding convergence is key to knowing how many terms are necessary to achieve a certain level of accuracy in representing the function.

Determining the smallest \(n\) for which the Fourier series yields \(|e_n(x)| \leq 0.01\) across the interval involves ensuring adequate convergence and gauging both the smoothness of the function and the behavior of its series representation.