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. $$ f(x)=\left\\{\begin{array}{ll}{0,} & {-\pi \leq x<-\pi / 2} \\ {1,} & {-\pi / 2 \leq x<\pi / 2} \\ {0,} & {\pi / 2 \leq x<\pi}\end{array}\right. $$

Short answer

#Answer# The Fourier series for the given function is: $$ f(x) = \frac{1}{2} + \sum_{k=0}^{\infty} \frac{4}{(2k+1)\pi}\cdot\cos((2k+1)x) , \ k=0,1,2,\dots $$ To sketch the graph of the function for three periods, plot the original function on the interval \(-\pi\) to \(\pi\) and repeat the pattern periodically on the intervals \(-3\pi\) to \(-\pi\) and \(\pi\) to \(3\pi\). The graph should extend from \(-3\pi\) to \(3\pi\).

Step-by-step solution

Find \(a_0\)

The first coefficient, \(a_0\), can be found by plugging the given function into the formula for \(a_n\) and evaluating the integral: $$ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx $$ Since the function is piecewise, we need to split the integral accordingly: $$ a_0 = \frac{1}{\pi} \left[\int_{-\pi}^{-\pi/2} 0 \, dx + \int_{-\pi/2}^{\pi/2} 1 \, dx + \int_{\pi/2}^{\pi} 0 \, dx\right] $$ $$ a_0 = \frac{1}{\pi} \left[0 + (\pi/2 - (-\pi/2)) + 0\right] = \frac{1}{\pi} \cdot \pi = 1 $$

Find the coefficients \(a_n\)

Using the formula for \(a_n\): $$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx $$ Again, we need to split the integral according to the given function: $$ a_n = \frac{1}{\pi} \left[\int_{-\pi}^{-\pi/2} 0 \cdot \cos(nx) \, dx + \int_{-\pi/2}^{\pi/2} 1 \cdot \cos(nx) \, dx + \int_{\pi/2}^{\pi} 0 \cdot \cos(nx) \, dx\right] $$ $$ a_n = \frac{1}{\pi} \int_{-\pi/2}^{\pi/2} \cos(nx) \, dx $$ After integrating and simplifying: $$ a_n = \frac{1}{\pi} \left[\frac{1}{n}\sin(nx) \right]_{-\pi/2}^{\pi/2} = \frac{2}{n\pi}(\sin(n\pi/2) - \sin(-n\pi/2)) $$ Since \(a_n\) is odd (its denominator \(n\) is odd), all even values of \(n\) lead to \(0\), and we get: $$ a_n = \left\{ \begin{array}{ll} 0 \,, & \mathrm{if} \ n=2k \,, \ k=1,2,3,\dots \\ \\ \displaystyle \frac{4}{n\pi} \,, & \mathrm{if} \ n=2k+1 \,, \ k=0,1,2,\dots \end{array} \right. $$

Find the coefficients \(b_n\)

Using the formula for \(b_n\): $$ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx $$ Because \(f(x)\) is an even function and sine is an odd function, the product \(f(x)\cdot \sin(nx)\) is odd. Thus, the integral from \(-\pi\) to \(\pi\) will be zero: $$ b_n = 0 $$

Construct the Fourier Series

Now that we have all the coefficients, we can construct the Fourier series: $$ f(x) = \frac{1}{2} + \sum_{k=0}^{\infty} \frac{4}{(2k+1)\pi}\cdot\cos((2k+1)x) , \ k=0,1,2,\dots $$

Sketch the Graph

To sketch the graph of the function to which the series converges for three periods, plot the original function \(f(x)\) on the given interval (from \(-\pi\) to \(\pi\)), and then repeat this pattern periodically for the other periods. In this case, we need to sketch three periods, so the graph should extend from \(-3\pi\) to \(3\pi\). Use the Fourier series we constructed in Step 4 to visually show the approximation on those three periods. Keep in mind that the convergence of this series is better at the points of the function different from the discontinuities.

Key concepts

Piecewise Function

A piecewise function is a type of function defined by different expressions based on the input value. Essentially, the function's rule changes depending on the "piece" or interval the input belongs to. For example, consider the function \( f(x) \) given in the exercise:

• From \(-\pi \leq x < -\pi/2\), \( f(x) = 0 \)
• From \(-\pi/2 \leq x < \pi/2\), \( f(x) = 1 \)
• From \(\pi/2 \leq x < \pi\), \( f(x) = 0 \)

This setup helps in designing functions that behave differently over specific intervals. They are crucial in modeling real-world situations where a single formula is inadequate.

Periodically Extended Function

A periodically extended function repeats its behavior over intervals of a fixed length, known as the period. In the exercise above, the function is extended periodically outside its initial interval \(-\pi\) to \(\pi\). By doing so, it creates a continuous wave pattern that can be analyzed using Fourier series.
Periodic extension allows the original behavior of the function to be repeated over an infinite domain, making it possible to describe phenomena like sound waves and signal processing. When a function is periodic, it has consistent patterns, making it easier to handle mathematically.

Trigonometric Series

A trigonometric series is a mathematical series composed of sine and cosine functions. The Fourier series, a special type of trigonometric series, breaks down periodic functions into sums of sines and cosines. Given our function, its Fourier series was built by determining coefficients \(a_n\), \(b_n\), and \(a_0\), based on integrals over one period.

The resulting series approximates the original function, capturing both its periodicity and shape. The formula found in the exercise:

• Includes only cosine terms because the sine coefficients \(b_n\) disappeared (since the function was even).
• Assists in constructing the series that converges to the behavior of the original function.

Utilizing this tool, complex signals can be represented more simply, aiding in analysis and problem-solving.

Integration of Trigonometric Functions

Integration of trigonometric functions involves calculating the area under curves defined by these functions. It is essential for determining the coefficients in a Fourier series.

For example, when finding \(a_n\) and \(b_n\), the integration of \( f(x)\cos(nx) \) and \( f(x)\sin(nx) \) over one period occurs. This process allows us to weigh how much of each trigonometric component is present in the function.

Key points to remember:

• Cosine integrals contribute to the \(a_n\) coefficients.
• Sine integrals contribute to the \(b_n\) coefficients.
• Even functions lead to zero \(b_n\) coefficients, simplifying calculations.

Integration skills are vital in applying Fourier series effectively, enabling the transformation of functions into sums of basic trigonometric expressions.