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}{cl} x / 2, & 0

Short answer

The period is 3. The Fourier series is a combination of sine and cosine functions with the calculated coefficients.

Step-by-step solution

Define the Period

Identify the length of one period of the given function. Here, the function is defined from 0 to 3, so the period T is 3.

Extend the Function for Several Periods

Sketch the function over several periods by repeating the interval from 0 to 3.

Express the Function in Terms of Sine and Cosine

The Fourier series of a periodic function with period T is given by: \[ f(x) = a_0 + \sum_{n=1}^{fty} \left( a_n \cos\left( \frac{2\pi n x}{T} \right) + b_n \sin\left( \frac{2\pi n x}{T} \right) \right) \] where \[ a_0 = \frac{1}{T} \int_{0}^{T} f(x) \, dx \] \[ a_n = \frac{2}{T} \int_{0}^{T} f(x) \cos\left( \frac{2\pi n x}{T} \right) \, dx \] \[ b_n = \frac{2}{T} \int_{0}^{T} f(x) \sin\left( \frac{2\pi n x}{T} \right) \, dx \]

Compute the Coefficient a_0

Calculate the average value of the function over one period. \[ a_0 = \frac{1}{T} \int_{0}^{3} f(x) \, dx \} = \frac{1}{3} \left( \int_{0}^{2} \frac{x}{2} \, dx + \int_{2}^{3} 1 \, dx \right) = \frac{1}{3} \left( \frac{1}{4} \left[ x^2 \right]_0^2 + \left[ x \right]_2^3 \right) = \frac{1}{3} \left( \frac{1}{4} (4 - 0) + (3 - 2) \right) = \frac{1}{3} (1 + 1) = \frac{2}{3} \]

Compute the Coefficients a_n

Use the formula for \( a_n \) to find the coefficients of the cosine terms. \[ a_n = \frac{2}{3} \int_{0}^{3} f(x) \cos\left( \frac{2\pi n x}{3} \right) \, dx = \frac{2}{3} \left( \int_{0}^{2} \frac{x}{2} \cos\left( \frac{2\pi n x}{3} \right) \, dx + \int_{2}^{3} 1 \cos\left( \frac{2\pi n x}{3} \right) \, dx \right) \]

Compute the Coefficients b_n

Use the formula for \( b_n \) to find the coefficients of the sine terms. \[ b_n = \frac{2}{3} \int_{0}^{3} f(x) \sin\left( \frac{2\pi n x}{3} \right) \, dx = \frac{2}{3} \left( \int_{0}^{2} \frac{x}{2} \sin\left( \frac{2\pi n x}{3} \right) \, dx + \int_{2}^{3} 1 \sin\left( \frac{2\pi n x}{3} \right) \, dx \right) \]

Write the Fourier Series

Combine all calculated coefficients to write the complete Fourier series. \[ f(x) =\frac{2}{3} + \sum_{n=1}^{fty} \left( a_n \cos\left( \frac{2\pi n x}{3} \right) + b_n \sin\left( \frac{2\pi n x}{3} \right) \right) \]

Key concepts

Periodic Functions

A periodic function is one that repeats its values in regular intervals or periods. For the given function in the exercise, the period is between 0 and 3. This means the function repeats every 3 units along the x-axis. Understanding periodic functions is essential because they allow us to model and analyze repetitive phenomena such as sound waves, seasons, and electrical signals. In Fourier series, we focus on decomposing these periodic functions into simpler trigonometric functions (sine and cosine).

To visualize a periodic function:

• Sketch several periods by extending the base interval. For instance, repeat the function from 0 to 3 over multiple intervals like -3 to 0 and 3 to 6.
• Observe that the pattern remains consistent across all intervals.

This extension helps grasp the concept of periodicity and is the first step in analyzing and expanding the function using a Fourier series.

Fourier Coefficients

Fourier coefficients are the constants we need to compute in order to express a periodic function as a sum of sine and cosine functions. These coefficients ( a_n and b_n ) quantify the contribution of each sine and cosine function in the series.

The general Fourier series formula is:
f(x) = a_0 + ∑_{n=1}^{∞} ( a_n cos( 2πnx/T ) + b_n sin( 2πnx/T ) ) Here, a_0 is the average value of the function over one period, calculated as:
a_0 = 1/T ∫_0^T f(x) dx The cosine coefficients ( a_n ) and sine coefficients ( b_n ) are found using:
a_n = 2/T ∫_0^T f(x) cos( 2πnx/T ) dx b_n = 2/T ∫_0^T f(x) sin( 2πnx/T ) dx
These integrals involve multiplying the function by sine or cosine and averaging over one period. Each coefficient reveals how much of that specific sine or cosine wave is in the function.

Sine and Cosine Functions

Sine and cosine functions are foundational in the study of periodic functions and Fourier series. They describe smooth, wave-like phenomena that repeat at regular intervals.

• Cosine function: It starts at its maximum value when x = 0 and oscillates between -1 and 1. It is an even function, meaning cos(x) = cos(-x) .
• Sine function: It starts at 0 when x = 0 and also oscillates between -1 and 1. It is an odd function, meaning sin(x) = -sin(-x) .

When breaking down complex periodic functions into their basic components using Fourier series, sine and cosine functions serve as the building blocks. They help in representing any periodic shape through a combination of these basic waves.
In practical terms:

• Cosine terms ( a_n cos( 2πnx/T )) contribute to the even symmetry parts of the function.
• Sine terms ( b_n sin( 2πnx/T )) contribute to the odd symmetry parts of the function.

Understanding these properties helps in both setting up the integrals needed to find Fourier coefficients and in conceptualizing the overall shape of the expanded series.