Question

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series. $$f(x)=\left\\{\begin{array}{ll}x / 2, & 0< x<2 \\\1, & 2< x<3\end{array}\right.$$.

Short answer

The function repeats every 3 units. Its Fourier series includes coefficients calculated as averages over each period.

Step-by-step solution

Define the Period of the Function

Identify the period of the given function. From the problem statement, the function is defined over one period from 0 to 3. Thus, the period is 3.

Sketch the Function Over One Period

Sketch the function on the interval from 0 to 3. From 0 to 2, the function is defined as \( f(x) = \frac{x}{2} \) and from 2 to 3, the function is defined as \( f(x) = 1 \).

Extend the Sketch to Multiple Periods

Replicate the function sketch for several periods. Since the period is 3, repeat the given function for intervals [3, 6], [6, 9], etc. This helps visualize the periodic nature of the function.

Compute Fourier Coefficients

The Fourier series for a function with period \( T \) is given by: \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos \frac{2\pi nx}{T} + b_n \sin \frac{2\pi nx}{T} \right) \]To compute the coefficients, use the formulas:\[ a_0 = \frac{2}{T} \int_0^T f(x) \, dx \]\[ a_n = \frac{2}{T} \int_0^T f(x) \cos \frac{2\pi nx}{T} \, dx \]\[ b_n = \frac{2}{T} \int_0^T f(x) \sin \frac{2\pi nx}{T} \, dx \]First, calculate \( a_0 \). For \( 0 < x < 2 \), \( f(x) = \frac{x}{2} \), and for \( 2 < x < 3 \), \( f(x) = 1 \).

Calculate \( a_0 \)

Compute \( a_0 \):\[ a_0 = \frac{2}{3} \left( \int_0^2 \frac{x}{2} \, dx + \int_2^3 1 \, dx \right) = \frac{2}{3} \left( \left[ \frac{x^2}{4} \right]_0^2 + \left[ x \right]_2^3 \right) = \frac{2}{3} \left( 1 + 1 \right) = \frac{4}{3} \]

Calculate \( a_n \) for \( n \geq 1 \)

Compute \( a_n \):\[ a_n = \frac{2}{3} \left( \int_0^2 \frac{x}{2} \, \cos \frac{2\pi nx}{3} \, dx + \int_2^3 1 \, \cos \frac{2\pi nx}{3} \, dx \right) \]Evaluate each integral separately.

Calculate \( b_n \) for \( n \geq 1 \)

Compute \( b_n \):\[ b_n = \frac{2}{3} \left( \int_0^2 \frac{x}{2} \, \sin \frac{2\pi nx}{3} \, dx + \int_2^3 1 \, \sin \frac{2\pi nx}{3} \, dx \right) \]Evaluate each integral separately.

Construct the Fourier Series

Combine the coefficients into the Fourier series formula. Use the computed values of \( a_0 \), \( a_n \) and \( b_n \) to write the full Fourier expansion of the function.

Key concepts

Periodic Functions

Periodic functions repeat their values at regular intervals. The interval over which the function repeats is called the period. For example, the given function in the exercise has a period of 3. This means every 3 units along the x-axis, the function values repeat.
Simplifying the analysis of periodic functions using Fourier series helps in various applications such as signal processing and heat distribution studies. When sketching periodic functions, after plotting over one period, extend the plot by replicating the pattern over subsequent intervals. This visual repetition helps in identifying the behavior and formulating the Fourier series easily.

Fourier Coefficients

Fourier coefficients help decompose a periodic function into sums of sines and cosines. These coefficients, denoted by \(a_0\), \(a_n\), and \(b_n\), are computed using integral calculus.

• The coefficient \(a_0\) represents the average value of the function over one period.


• The coefficients \(a_n\) and \(b_n\) quantify the contributions of cosine and sine functions respectively at different frequencies to the overall function.


• Their formulas are:

\[ a_0 = \frac{2}{T} \int_0^T f(x) \, dx \] \[ a_n = \frac{2}{T} \int_0^T f(x) \cos \frac{2\pi nx}{T} \, dx \] \[ b_n = \frac{2}{T} \int_0^T f(x) \sin \frac{2\pi nx}{T} \, dx \]
Calculating these integrals domain-wise (as shown in steps 4 to 7 of the exercise) accurately gives the required Fourier coefficients.

Integral Calculus

Integral calculus is used extensively in finding Fourier coefficients. The process involves integrating the function over one period. For any function f(x), the use of integrals allows measuring the area under the curve, which is crucial for calculating \(a_0\), \(a_n\), and \(b_n\).
Properly evaluating these integrals requires careful handling of the function's different segments, if it is piece-wise defined. In the given exercise:

• \(f(x) = \frac{x}{2} \) from 0 to 2


• \(f(x) = 1 \) from 2 to 3.

The calculated integrals such as \[ a_0 = \frac{2}{T} \int_0^2 \frac{x}{2} \, dx + \int_2^3 1 \, dx = \frac{4}{3} \] help derive the Fourier series representation.

Function Sketching

Sketching the function is the preliminary step in understanding the problem and eventually extending it to multiple periods. Start by plotting it over the given period first. For the function in the exercise, divide it as:

• From 0 to 2, sketch the linear part \(f(x) = \frac{x}{2} \)


• From 2 to 3, plot the constant \(f(x) = 1 \).

Extend the sketch by copying the pattern over several periods: 3 to 6, 6 to 9, etc. This repetition demonstrates the periodic behavior clearly and helps in verifying the Fourier series represents the original function accurately.
Accurate function sketching not only aids in visualizing the function but also simplifies the computation process for Fourier coefficients.