Question

Given \(f(x)=|x|\) on \((-\pi, \pi),\) expand \(f(x)\) in an appropriate Fourier series of period 2 \(\pi.\)

Short answer

The Fourier series for

Step-by-step solution

Identify the Fourier Series Form

The function can be expanded in a Fourier series as: where we need to identify the coefficients.

Compute the Fourier Coefficients

To find the Fourier coefficients, we'll calculate the following integrals:

Calculate the Constant Term

The term is given by:

Calculate the Cosine Coefficients

The

Calculate the Sine Coefficients

Since

Construct the Fourier Series

The Fourier series expansion becomes:

Key concepts

Fourier coefficients

In the Fourier series, coefficients play a crucial role. They determine how much of each sine and cosine wave is present in the overall function.

To find these coefficients for a function like \(f(x) = |x|\), we use integrals. There are three types of coefficients to find:

• The constant term \(a_0\)
• The cosine coefficients \(a_n\)
• The sine coefficients \(b_n\)

For the given function on the interval \( \(-\pi\), \pi\right)\), these integrals are calculated with the following formulas:

The constant term:
\[ a_0 = \frac{1} {2\pi} \int_{{-\pi}}^{{\pi}} f(x) dx \]

The cosine coefficients:
\[ a_n = \frac{1} {\pi} \int_{{-\pi}}^{{\pi}} f(x) \cos(nx) dx \]

The sine coefficients:
\[ b_n = \frac{1} {\pi} \int_{{-\pi}}^{{\pi}} f(x) \sin(nx) dx \]

These coefficients help build the final Fourier series, giving us a powerful way to represent periodic functions.

Absolute value function

The absolute value function, denoted by \( |x| \), is a very common mathematical function. It measures the distance of a number from zero on the number line.

It's defined as follows:

• \( |x| = x \) if \( x \geq 0 \)
• \( |x| = -x \) if \( x < 0 \)

For example, \( |3| = 3 \) and \( |-3| = 3 \). This function is particularly interesting because it is non-linear and has a 'V' shape.

In the case of a Fourier series on the interval \( \(-\pi\), \pi\right)\), the absolute value function \(|x|\) is symmetrical and periodic. This symmetry is important when calculating the Fourier coefficients.

Periodic functions

A function is called periodic if it repeats itself after a fixed interval, known as the period. Mathematically, a function \(f(x)\) is periodic with period \(T\) if:
\[ f(x + T) = f(x) \]

• Sine and cosine functions are classic examples of periodic functions with period \(2\pi\).
• The function \(f(x) = |x|\) is also periodic with period \(2\pi\) on the given interval.

Periodic functions are crucial in Fourier analysis because they allow the function to be represented as a sum of sines and cosines. This makes complex functions easier to study and understand.

In this exercise, the function \(f(x) = |x|\) is expanded in a Fourier series with period \(2\pi\), aligning perfectly with its periodicity.

Trigonometric series

A trigonometric series is a series of terms that are trigonometric functions (sine and cosine functions). The general form of a Fourier series is a trigonometric series:
\[ f(x) = a_0 + \sum_{{n=1}}^{{\infty}} \(a_n \cos(nx) + b_n \sin(nx) \) \]

This series combines both sine and cosine terms to approximate the original function. Each term in the series corresponds to a different frequency component of the original signal.

The coefficients \(a_n\) and \(b_n\) determine the amplitude of each cosine and sine term respectively. Calculating these coefficients accurately is essential to represent the function correctly.

In the given exercise, the absolute value function \(f(x) = |x|\) is expanded into a trigonometric series to approximate it accurately on the interval \( \(-\pi\), \pi\right)\). This approach is very powerful for analyzing & understanding the behavior of periodic functions.