Question

(a) Sketch the graph of the given function for three periods. (b) Find the Fourier series for the given function. $$ f(x)=\left\\{\begin{array}{lr}{x,} & {-\pi \leq x < 0,} \\ {0,} & {0 \leq x<\pi ;}\end{array} \quad f(x+2 \pi)=f(x)\right. $$

Short answer

Question: Sketch the graph of the function $$f(x) = \begin{cases} x, & -\pi \leq x < 0, \\ 0, & 0 \leq x < \pi. \end{cases}$$ for three periods and find its Fourier series. Answer: The Fourier series of $$f(x)$$ is given by $$f(x) \approx \frac{\pi}{4} - \sum_{n=1}^\infty \frac{1}{n^2\pi} (\cos{n\pi} - 1) \cos{(nx)}$$ with zero sine components, i.e., $$b_n = 0$$ for all $$n$$.

Step-by-step solution

Sketch the graph of the function for three periods

To sketch the graph of the function $$f(x)$$ for three periods, observe the definition of the function: $$ f(x) = \left\{ \begin{array}{lr} x, & -\pi \leq x < 0, \\ 0, & 0 \leq x < \pi. \end{array} \right. $$ The function is periodic with a period of $$2\pi$$. Notice that for each period: - The function is a diagonal line with slope equal to 1 for $$-\pi \leq x < 0$$ - The function equals 0 for $$0 \leq x < \pi$$ Sketch the graph by first drawing one period of the function from $$-\pi$$ to $$\pi$$. Then, extend this sketch to three periods by replicating the graph between $$-\pi$$ and $$\pi$$ two more times. The graph should cover the range $$[-3\pi, 3\pi]$$.

Calculate the Fourier coefficients

The Fourier series of a periodic function is given by: $$ f(x) \approx \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos{(\frac{2\pi nx}{2\pi})} + b_n \sin{(\frac{2\pi nx}{2\pi})} \right], $$ where $$ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx, $$ $$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos{(nx)} dx, $$ and $$ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin{(nx)} dx. $$

Calculate $$a_0$$

Compute $$a_0$$ using the formula and the definition of $$f(x)$$: $$ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx = \frac{1}{\pi} \left( \int_{-\pi}^0 x dx + \int_0^\pi 0 dx \right). $$ The second integral is 0, so we only need to compute the first integral: $$ a_0 = \frac{1}{\pi} \int_{-\pi}^0 x dx = \frac{1}{\pi} \left[ \frac{1}{2}x^2 \right]_{-\pi}^0 = \frac{\pi}{2}. $$

Calculate $$a_n$$

Compute $$a_n$$ using the formula and the definition of $$f(x)$$: $$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos{(nx)} dx = \frac{1}{\pi} \left( \int_{-\pi}^0 x \cos{(nx)} dx + \int_0^\pi 0 \cos{(nx)} dx \right). $$ Again, the second integral is 0, so compute the first integral: $$ a_n = \frac{1}{\pi} \int_{-\pi}^0 x \cos{(nx)} dx = \frac{1}{n\pi} \left[ x\sin{(nx)} \right]_{-\pi}^0 - \frac{1}{n\pi} \int_{-\pi}^0 \sin{(nx)} dx. $$ Evaluate the remaining integral: $$ a_n = -\frac{1}{n\pi} \left[ -\frac{1}{n}\cos{(-n\pi)}\right]_{-\pi}^0 = -\frac{1}{n^2\pi} (\cos{n\pi} - 1). $$

Calculate $$b_n$$

Compute $$b_n$$ similarly, using the definition of $$f(x)$$: $$ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin{(nx)} dx = \frac{1}{\pi} \left( \int_{-\pi}^0 x \sin{(nx)} dx + \int_0^\pi 0 \sin{(nx)} dx \right). $$ As before, the second integral is 0, so compute the first integral: $$ b_n = \frac{1}{\pi} \int_{-\pi}^0 x \sin{(nx)} dx = \frac{1}{n\pi} \left[-x\cos{(nx)} \right]_{-\pi}^0 - \frac{1}{n\pi} \int_{-\pi}^0 -\cos{(nx)} dx = -\frac{1}{n^2\pi} \sin{n\pi}. $$ Since $$\sin{n\pi} = 0$$ for all integer values of $$n$$, we have $$b_n = 0$$.

Write the Fourier series

Finally, write out the Fourier series by combining the calculated values of $$a_0$$, $$a_n$$, and $$b_n$$: $$ f(x) \approx \frac{\pi}{4} - \sum_{n=1}^\infty \frac{1}{n^2\pi} (\cos{n\pi} - 1) \cos{(nx)}. $$

Key concepts

Periodic Functions

In mathematics, a periodic function is a function that repeats its values at regular intervals or periods. These functions are significant because they describe phenomena that inherently repeat over time, such as sound waves or tides.
In this exercise, the function given is periodic with a period of \(2\pi\). This means the behavior of the function from \(-\pi\) to \(\pi\) repeats indefinitely. In essence, after every \(2\pi\) interval, the entire graph of the function can be duplicated to accurately depict its shape over the entire domain.

• The function is defined piecewise: it's a diagonal line with a slope of 1 from \(-\pi\) to 0, and it is zero from 0 to \(\pi\).
• Understanding the periodicity helps in predicting the function's behavior even beyond the initially given segment.

For effective analysis and calculations, recognizing periodicity allows us to work within a single period yet understand the function's behavior over its infinite span.

Fourier Coefficients

Fourier coefficients are crucial when expressing periodic functions as a series of sinusoidal components. These coefficients form the backbone of Fourier series, which decompose functions into sums of sines and cosines.
To determine the Fourier series of a function, we calculate the coefficients: \(a_0\), \(a_n\), and \(b_n\). These coefficients help reconstruct the function using trigonometric terms:

• The coefficient \(a_0\) represents the average value or DC component of the function. It is found by integrating the function over its full period.
• The terms \(a_n\) and \(b_n\) quantify the weights of the cosine and sine components at each harmonic frequency \(n\).

For the given function, these coefficients are calculated using integration over the period \([-\pi, \pi]\). Understanding how to compute these coefficients is essential, as they allow converting a complex periodic signal into manageable sinusoidal components.

Sketching Graphs

Sketching graphs of functions, especially periodic ones, involves understanding how to accurately depict their behavior over multiple cycles. The task is to translate the mathematical expression of a function into a visual representation.
The exercise involves sketching the given function across three periods, from \(-3\pi\) to \(3\pi\), using its piecewise definition:

• For \(-\pi \leq x < 0\), sketch a line with a slope of 1, creating a rising diagonal line starting from \(-\pi\) on the x-axis to 0.
• From 0 to \(\pi\), draw a flat line along the x-axis, since the function equals zero.

Repeat these steps to replicate the function's behavior across the specified range. Accurately sketching such graphs requires attention to detail, ensuring periodic features are correctly mirrored over each cycle. This skill is valuable in both mathematics and applied areas such as engineering and physics.

Integration Techniques

Integration is a mathematical technique used to find the accumulated area under curves or to determine the antiderivative of a function. It is a central tool when applying calculus to derive quantities like area, volume, and other accumulative properties.
In calculating Fourier coefficients, integration is used extensively:

• To calculate \(a_0\), the integral of the function over one period is used. This involves simple area calculation techniques when the function is straightforward.
• Finding \(a_n\) and \(b_n\) involves integrating the product of the function with trigonometric functions, which can be more complex. It involves careful substitution and utilization of standard integration methods.

Understanding integration processes and techniques ensures accurate computation of these coefficients. Mastery of integration is thus crucial in analyzing and decomposing complex functions into their Fourier series representation.