Question

In each of the following problems you are given a function on the interval \(-\pi

Short answer

The Fourier series expansion is: \[ f(x) = \frac{1}{4} + \sum_{n=1}^{\infty} \left( \frac{\sin(n\pi/2)}{n\pi}\cos(nx) + \frac{1 - \cos(n\pi/2)}{n\pi} \sin(nx) \right) \]\]

Step-by-step solution

Understand the Function and Its Intervals

The function provided is a piecewise function defined on the interval \(-\pi < x < \pi\). It has different values in different sub-intervals: \({0\) in \( -\pi < x < 0\), \(1\) in \(0 < x < \frac{\pi}{2}\), and \(0\) in \( \frac{\pi}{2} < x < \pi\).

Sketch Several Periods of the Function

Since the function is periodic with period \(2\pi\), extend the given function to other intervals by repeating the shape. Draw the function from \(-\pi < x < \pi\) and then repeat the same piecewise pattern to intervals like \(-3\pi < x < -\pi\) and \( \pi < x < 3\pi\). Ensure that the pieces connect properly at the boundaries \(-\pi\) and \( \pi\).

Write Down the General Form of the Fourier Series

The Fourier series expansion for a function with period \(2\pi\) is given by:\[ f(x) = a_0 + \sum_{n=1}^{\infty} \left(a_n \cos(nx) + b_n \sin(nx)\right) \]

Calculate the Fourier Coefficients

Compute the coefficients \(a_0, a_n, \) and \(b_n \) using the following integrals: \[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx \]\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \]\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \]

Compute \(a_0\)

Use the definition to find \(a_0\): \[a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx = \frac{1}{2\pi} \left( \int_{-\pi}^{0} 0 \, dx + \int_{0}^{\frac{\pi}{2}} 1 \, dx + \int_{\frac{\pi}{2}}^{\pi} 0 \, dx\right)\]\[a_0 = \frac{1}{2\pi} \left( 0 + \int_{0}^{\frac{\pi}{2}} 1 \, dx + 0 \right) = \frac{1}{2\pi} \cdot \frac{\pi}{2} = \frac{1}{4}\]

Compute \(a_n\)

Use the definition to find \(a_n\):\[a_n = \frac{1}{\pi} \int_{-\pi}^{0} 0 \cos(nx) \, dx + \frac{1}{\pi} \int_{0}^{\frac{\pi}{2}} \cos(nx) \, dx+ \frac{1}{\pi} \int_{\frac{\pi}{2}}^{\pi} 0 \cos(nx) \, dx \]\[ a_n = \frac{1}{\pi} \left( 0 + \int_{0}^{\frac{\pi}{2}} \cos(nx) \ dx + 0 \right) = \frac{1}{\pi} \left[ \frac{\sin(nx)}{n} \right]_0^{\frac{\pi}{2}}\]\[= \frac{1}{\pi} \left( \frac{\sin(n\pi/2)}{n} - \frac{\sin(0)}{n} \right) = \frac{1}{\pi} \cdot \frac{\sin(n\pi/2)}{n}\]\[ = \frac{\sin(n\pi/2)}{n\pi} \]

Compute \(b_n\)

Use the definition to find \(b_n\):\[ b_n = \frac{1}{\pi} \int_{-\pi}^{0} 0 \sin(nx) \, dx + \frac{1}{\pi} \int_{0}^{\frac{\pi}{2}}\sin(nx) \, dx+ \frac{1}{\pi} \int_{\frac{\pi}{2}}^{\pi} 0 \sin(nx) \, dx \]\[ b_n = \frac{1}{\pi} \left( 0 + \int_{0}^{\frac{\pi}{2}} \sin(nx) \, dx + 0 \right) = \frac{1}{\pi} \left[ -\frac{\cos(nx)}{n} \right]_0^{\frac{\pi}{2}} \]\[ b_n = \frac{1}{\pi} \left( -\frac{\cos(n\pi/2)}{n} + \frac{\cos(0)}{n} \right) = \frac{1}{\pi} \cdot \frac{-\cos(n\pi/2) + 1}{n}\ \]\[= \frac{1 - \cos(n\pi/2)}{n\pi} \]

Put It All Together

Substitute \(a_0, a_n, b_n\) into the Fourier series form:\[ f(x) = a_0 + \sum_{n=1}^{\infty} \left(a_n \cos(nx) + b_n \sin(nx)\right) \]\[ = \frac{1}{4} + \sum_{n=1}^{\infty} \left( \frac{\sin(n\pi/2)}{n\pi} \cos(nx) + \frac{1 - \cos(n\pi/2)}{n\pi} \sin(nx) \right) \]

Key concepts

Piecewise Function

A piecewise function is a function composed of multiple sub-functions, each with its own domain. In this exercise, the function has three parts, each defined over a specific interval:

• For \(-\pi < x < 0\), the function is equal to 0.
• For \(0 < x < \frac{\pi}{2}\), the function is equal to 1.
• For \(\frac{\pi}{2} < x < \pi\), the function is back to 0.

These parts are stitched together, making the function change its output based on the interval in which the input falls. Understanding piecewise functions is crucial in constructing their Fourier series expansions.

Periodic Function

A periodic function repeats its values in regular intervals or periods. For this exercise, the given function is periodic with a period of \(2\pi\). This means that the function pattern from \(-\pi < x < \pi\) is replicated over every interval of length \(2\pi\). Sketching several periods helps visualize how the function extends beyond the initial interval. It involves repeating the piecewise segments over adjoining intervals such as \(-3\pi < x < -\pi\) and \(\pi < x < 3\pi\). This repetition helps in understanding the global behavior of the periodic function.

Fourier Coefficients

Fourier coefficients are essential in breaking down a periodic function into its sine and cosine components. For a function with period \(2\pi\), the Fourier series coefficients \(a_0\), \(a_n\), and \(b_n\) are calculated as follows:

• \(a_0\): This is the average value of the function over one period. It is calculated using the integral \(a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx\).
• \(a_n\): These coefficients for cosine terms are found using \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx\).
• \(b_n\): These coefficients for sine terms are calculated using \(b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx\).

Solving these integrals accurately breaks down the function into its sine and cosine series components, enabling the Fourier series representation.

Sine-Cosine Series

The sine-cosine series form is a representation of a periodic function as an infinite sum of sines and cosines. The general form of the Fourier series for a function with period \(2\pi\) is:\[ f(x) = a_0 + \sum_{n=1}^{\infty} \left(a_n \cos(nx) + b_n \sin(nx)\right) \]Here,

• \(a_0\) represents the average (DC) component of the function.
• \(a_n\) are the coefficients of the cosine terms which capture the even symmetry of the function.
• \(b_n\) are the coefficients of the sine terms which capture the odd symmetry of the function.

This decomposition helps in understanding and analyzing the behavior of the function in terms of its frequency components.

Intervals

Intervals are the specific ranges over which parts of the piecewise function are defined. In the given function, there are three distinct intervals:

• From \(-\pi\) to 0,
• From 0 to \(\frac{\pi}{2}\),
• From \(\frac{\pi}{2}\) to \(\pi\).

Each interval specifies a different value of the function, making it crucial to understand how these intervals contribute to the whole piecewise definition. Additionally, when extending the function for periodic repetition, these intervals are key in accurately drawing the function across a broader range.