Question

Find the required Fourier series for the given function and sketch the graph of the function to which the series converges over three periods. $$ \begin{array}{l}{f(x)=L-x, \quad 0 \leq x \leq L ; \quad \text { cosine series, period } 2 L} \\ {\text { Compare with Example } 1 \text { of Section } 10.2 .}\end{array} $$

Short answer

Question: Determine the Fourier cosine series for the function f(x) = L - x in the interval [0, L] with the period 2L. Sketch the graph of this function converging to f(x) over three periods. Answer: The Fourier cosine series for the function f(x) = L - x is given by: $$ f(x) = L + \sum_{n=1}^{\infty} \frac{2L}{(n\pi)^2}(-1)^n \cos{\frac{n \pi x}{L}} $$ To sketch the graph of this function over three periods [-2L, 4L], you would first draw a sawtooth wave that decreases from L to 0 in each interval of length L, and then oscillate near these points due to the cosine terms in the Fourier series. This approximation will improve as you add more terms to the series, and the graph will show the Gibbs phenomenon near the jump discontinuities as it oscillates and converges to the function f(x) = L-x in the interval [0, L].

Step-by-step solution

Determine the Fourier cosine series coefficients

According to the definition of the Fourier cosine series, we have the Fourier series as: $$ f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos{\frac{n \pi x}{L}} $$ We can find the coefficients \(a_0\) and \(a_n\) using the following equations: $$ a_0 = \frac{2}{L} \int_{0}^{L} f(x) dx \\ a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos{\frac{n \pi x}{L}} dx $$

Calculate the a_0 coefficient

Plugging f(x) into the equation for \(a_0\), we get: $$ a_0 = \frac{2}{L} \int_{0}^{L} (L-x) dx $$ Now, integrate and simplify to find \(a_0\): $$ a_0 = \frac{2}{L}\Big[ Lx - \frac{1}{2}x^2 \Big]_{0}^{L} = \frac{2}{L}\Big[ \frac{1}{2}L^2\Big] = L $$

Calculate the a_n coefficients

Next, plug f(x) into the equation for \(a_n\) to solve for the \(a_n\) coefficients: $$ a_n = \frac{2}{L} \int_{0}^{L} (L-x) \cos{\frac{n \pi x}{L}} dx $$ Now, we can apply integration by parts with \(u = L-x\) and \(dv = \cos(\frac{n \pi x}{L})dx\) to find \(a_n\): $$ \begin{aligned} a_n &= \frac{2}{L} \left[ (L-x)\frac{L}{n\pi} \sin \frac{n \pi x}{L} \Big|_{0}^{L} - \int_{0}^{L} \frac{L}{n\pi} \sin \frac{n \pi x}{L} dx \right] \\ &= - \frac{2}{L} \int_{0}^{L} \frac{L}{n\pi} \sin \frac{n \pi x}{L} dx \end{aligned} $$ Now we can integrate and simplify to find \(a_n\): $$ a_n = - \frac{2L}{(n\pi)^2}(-1)^n = \frac{2L}{(n\pi)^2}(-1)^n $$

Formulate the Fourier cosine series

Using the coefficients \(a_0\) and \(a_n\) derived above, we can now write the Fourier cosine series for the function \(f(x)=L-x\) as: $$ f(x) = L + \sum_{n=1}^{\infty} \frac{2L}{(n\pi)^2}(-1)^n \cos{\frac{n \pi x}{L}} $$

Sketch the graph over three periods

With the Fourier cosine series in-hand, we can now sketch the graph of the function converging to f(x) = L-x over three periods. Since the period is 2L, we will sketch the graph from x = -2L to x = 4L. The graph will resemble a sawtooth wave, decreasing from L to 0 in each interval of length L, and repeating for each period. Due to the cosine terms in the Fourier series, the function will oscillate near these points, converging to the function f(x) = L-x on the given interval of [0, L]. Remember that the more terms added in the Fourier series, the better the approximation of the function will be, and the graph will show the characteristic Gibbs phenomenon near the jump discontinuities as it oscillates and converges to the given function.

Key concepts

Cosine series

Cosine series are part of Fourier series, which allows us to express a given periodic function as a sum of cosine functions with different frequencies. These series are especially useful for even functions, which are symmetrical around the y-axis. The general form of a cosine series is given by:

• \[ f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos{\frac{n \pi x}{L}} \]

where \( a_0 \) and \( a_n \) are the coefficients calculated for the specific function. By expressing a function in terms of its cosine components, you can approximate the original function over its period.
Different frequencies in the cosine terms contribute to the overall shape of the waveform, capturing both the sharp jumps and the gradual inclines.
In the exercise above, the aim is to find the cosine series for the linear function \( f(x) = L - x \). This series will help to model and understand complex periodic behaviors, like the repetitive sawtooth pattern shown in the sketching step.
Cosine series are widely applied in signal processing, physics, and engineering, where understanding and manipulating periodic functions is essential.

Integration by parts

Integration by parts is a useful technique in calculus for simplifying integrals that are products of functions. This method is derived from the product rule for differentiation. The formula for integration by parts is:

• \[ \int u \, dv = uv - \int v \, du \]

where \( u \) and \( v \) are differentiable functions. This technique was applied in the solution when calculating the Fourier coefficients, particularly \(a_n\).
In the example, \( u \) is chosen as \( L-x \) and \( dv \) as \( \cos{\frac{n \pi x}{L}} dx \). Applying integration by parts helps to manage and solve the integral of these products, leading to expressions that are easier to evaluate.
When applying integration by parts, strategically choosing \( u \) and \( dv \) can simplify the problem significantly. Ideally, you pick \( u \) as a function whose derivative simplifies the integral, and \( dv \) as a function whose integral is straightforward.
Using this technique is beneficial when working with Fourier series problems, as it often allows us to express terms in solvable forms and evaluate them efficiently.

Periodic functions

Periodic functions are functions that repeat their values in regular intervals or periods. They are characterized by the smallest positive interval \( T \) such that for all x, \( f(x + T) = f(x) \).
Common examples include sine and cosine functions, which naturally have periodic behavior due to their intrinsic trigonometric properties.

• The period \( L \) of the function given in the exercise is crucial as it determines the interval length over which the behavior of the function is analyzed and the series is summed.

In the exercise, the function \( f(x) = L - x \) is defined over the interval \( [0, L] \) and repeats every \( 2L \). This characteristic is what allows it to be expressed as a Fourier series, which decomposes the function into periodic sine and cosine components, rooted in the trigonometric identities.
Understanding periodic functions is vital in many areas, such as signal processing, where they model oscillating signals, and in physics, where they represent wave phenomena. In sketching their graph, one can expect repeated patterns that can be visually predicted and mathematically characterized over the specified interval, providing a deeper insight into the behavior of oscillatory systems.