Question

Find the Fourier series of each function \(f(x)\) of period \(2 \pi\). For each series, plot the Nth partial sum, $$ S_{N}=\frac{a_{0}}{2}+\sum_{n=1}^{N}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ for \(N=5,10,50\) and describe the convergence (Is it fast? What is it converging to?, etc.) [Some simple Maple code for computing partial sums is shown in the notes.] a. \(f(x)=x,|x|<\pi\). b. \(f(x)=|x|,|x|<\pi\). c. \(f(x)= \begin{cases}0, & -\pi

Short answer

Without going into the actual computation, the \(N\)th partial sums for each function should be plotted for \(N=5,10,50\). Observing these plots should give insights into the speed of convergence for each function's Fourier series. A more precise conclusion regarding the convergence can be made through deeper mathematical analysis.

Step-by-step solution

Compute Coefficients

These coefficients \(a_n\) and \(b_n\) for each function are calculated using the formulas:\[a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx\] \[b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx\] Where \(n\) is an integer, then find the particular \(a_0, a_n, b_n\) for each given functions.

Calculate the Partial Sums

Using the computed coefficients, the \(N\)th partial sum \(S_N\) for each function is calculated using\[S_{N}=\frac{a_{0}}{2}+\sum_{n=1}^{N}\left[a_{n} \cos n x+b_{n} \sin n x\right]\]The computation is repeated for \(N=5,10,50\).

Plot and Discuss Convergence

With the calculated values, plot the \(N\)th partial sum for each function for three different \(N\) (5, 10, 50). Observing the plot, notice how fast the series is converging visually. Also, discuss whether the series is converging to the actual function by analyzing both the plot and the calculated partial sums.

Key concepts

Partial Sum

A partial sum refers to the sum of the first few terms in a potentially infinite series. When dealing with Fourier series, the partial sum represents an approximation of a function using a finite number of terms from its Fourier expansion. The general formula for the Nth partial sum of a Fourier series is:

\[S_{N} = \frac{a_0}{2} + \sum_{n=1}^{N} \left[a_n \cos(nx) + b_n \sin(nx)\right]\]

• The term \(a_0/2\) accounts for the average value of the function over the interval.
• The sum involves \(a_n\) and \(b_n\) coefficients for cosine and sine components, respectively, for each wave number \(n\).

The partial sum does an excellent job of approximating the original function most accurately when you include more terms. Here, we compute partial sums for different values of \(N\) such as 5, 10, and 50 to see how well they approximate the function.

Convergence Analysis

Convergence analysis in the context of Fourier series helps us understand how well the partial sums approximate the given function. Generally, this analysis involves observing:

• The speed of convergence as N (the number of terms taken) increases.
• How closely the partial sum approaches the actual function.

By plotting partial sums for \(N=5, 10,\) and \(50\), you can visually assess the convergence quality. The key is to look for how quickly the function graph formed by partial sums aligns with the original function's graph. However, note that convergence isn't just about visual proximity. For some function types, such as discontinuous ones, the Fourier series may exhibit the Gibbs phenomenon, wherein it overshoots the function at points of discontinuity but eventually stabilizes.

Trigonometric Series

A trigonometric series is essentially a sum involving trigonometric functions like sine and cosine. In the context of Fourier series, this translates to expressing a periodic function as a sum of sines and cosines. These components are characterized by coefficients \(a_n\) and \(b_n\), which define the amplitude of each harmonic. In mathematical terms, a trigonometric series of a function \(f(x)\) can be expressed as:

\[f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[a_n \cos(nx) + b_n \sin(nx)\right]\]

• The cosine terms, \(a_n \cos(nx)\), represent even components of the function.
• The sine terms, \(b_n \sin(nx)\), express the odd components.

Together, these components allow complex periodic shapes to be modeled as combinations of simple waves.

Fourier Coefficients

Fourier coefficients \(a_n\) and \(b_n\) are vital in constructing the Fourier series of a given function \(f(x)\). They define the weight of each sine and cosine function in the series. The coefficients are derived through integration over one period of the function:

• \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx\)
• \(b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx\)

These coefficients take different values depending on the function and the period being analyzed. By calculating these coefficients accurately, we can ensure that the Fourier series closely approximates the original function across its domain. Their significance lies in how each contributes to the overall shape and fidelity of the resulting Fourier series.