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. $$ f(x)=1, \quad 0 \leq x \leq \pi ; \quad \text { cosine series, period } 2 \pi $$

Short answer

Answer: The Fourier cosine series for the function f(x) = 1 on the interval 0 ≤ x ≤ π with a period of 2π is F(x) = 1/2.

Step-by-step solution

Set up the Fourier Cosine Series formula

The Fourier cosine series of a function f(x) with period 2π is given by: $$ F(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos (nx) $$ where the coefficients a_n are given by: $$ a_n = \frac{1}{\pi} \int_0^{\pi} f(x) \cos(nx) dx $$

Determine the a_0 coefficient

To find a_0, we need to compute the integral: $$ a_0 = \frac{1}{\pi} \int_0^{\pi} f(x) dx = \frac{1}{\pi} \int_0^{\pi} 1 dx $$ Evaluating the integral, we get: $$ a_0 = \frac{1}{\pi}(\pi - 0) = 1 $$

Determine the a_n coefficients

Now we need to compute the a_n coefficients by computing the integral for n ≥ 1: $$ a_n = \frac{1}{\pi} \int_0^{\pi} f(x) \cos(nx) dx = \frac{1}{\pi} \int_0^{\pi} 1 \cdot \cos(nx) dx $$ Evaluating the integral, we get: $$ a_n = \frac{1}{\pi} \left[\frac{\sin(nx)}{n}\right]_0^{\pi} = \frac{1}{\pi}\left(\frac{\sin(n\pi)}{n} - \frac{\sin(0)}{n}\right) = 0 \quad \text{(since sin(nπ) = 0)} $$

Write down the Fourier Cosine Series

Now that we have our coefficients a_0 and a_n, we can write the Fourier cosine series as follows: $$ F(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos (nx) = \frac{1}{2} + \sum_{n=1}^{\infty} 0 \cdot \cos (nx) = \frac{1}{2} $$

Sketch the graph over three periods

Since our Fourier series converges to a constant function, the graph of the function to which the series converges is a horizontal line at height 1/2. To sketch the graph over three periods, we can draw a horizontal line from -6π to 6π with the height of 1/2.

Key concepts

Fourier Series Convergence

Understanding the convergence of Fourier series is crucial to comprehending how these mathematical tools work. The Fourier series of a function is said to converge at a point if the sequence of partial sums of the series approaches a finite limit at that point.

In the context of cosine Fourier series, for functions that are piecewise continuous and have a limited number of discontinuities, the convergence is typically guaranteed. The series converges to the function value itself at points where the function is continuous. At points of discontinuity, however, the series converges to the average of the left- and right-hand limits.

This is known as the Dirichlet's condition for convergence. From this perspective, we can see how the series for the given function in our problem, being a constant function within the specified domain, results in a Fourier series that converges to the constant function itself, in this case, to the constant value of 1/2 across every point within the designated interval.

Cosine Series Coefficients

The coefficients in a cosine Fourier series play a critical role in determining the shape and behavior of the function that the series represents. In a Fourier cosine series, we calculate these coefficients, denoted as a_n, using a specific integral formula.

The zeroth coefficient, a₀, dictates the baseline level or the average value of the function. In the problem provided, this coefficient was calculated to be 1, which establishes the constant function value of 1/2 after being halved as part of the series formulation.

The rest of the cosine series coefficients, a_n for n ≥ 1, determine the amplitude and phase of the cosine functions that make up the series. However, for our example, we found that all a_n are zero, meaning the resulting Fourier series lacks any additional cosine terms beyond the constant component provided by a₀. Essentially, the higher-order coefficients disappear because the integral of cosine multiplied by a constant over its period is zero, accentuating the series' convergence to a constant function.

Graph Sketching in Mathematics

Graph sketching is a vital skill in mathematics that helps visualize functions and their behavior. When it comes to Fourier series, sketching the graph of the function that the series converges to over several periods can be particularly illuminating.

In our exercise, sketching the graph over three periods is simplified due to the constant nature of the function. It's a horizontal line at the height of 1/2, reflecting the constant component of the Fourier series we calculated. Over the interval from -6π to 6π, the line does not vary in height, corresponding to the zero cosine coefficients a_n for n ≥ 1.

Importance of Scale and Periods

When sketching the graph of a Fourier series, it is important to accurately represent the scale for both the x-axis (the independent variable) and the y-axis (the dependent variable). Additionally, showing multiple periods—as done in the exercise when extending the graph from -6π to 6π—helps emphasize the periodic nature of the function and the regions of convergence.