Question

You are given \(f(x)\) on an interval, say \(0

Short answer

The even function is symmetric and periodic, the odd function is antisymmetric and periodic, and the function with period b repeats every b units. Fourier series for each are computed with appropriate coefficients.

Step-by-step solution

Sketching the Even Function

Given the function defined on \(0 < x < 3\): $$ f(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \end{cases} $$ First, extend this function to be even with a period of \(2b\) where \(b=3\). Create a mirror image of the function in the interval \(-3 < x < 0\), and repeat it periodically. This gives: $$ f_c(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \ 1 & -1 < x < 0 \ 0 & -3 < x < -1 \end{cases} $$

Sketching the Odd Function

Next, extend this function to be odd with a period of \(2b\). In the interval \(-3 < x < 0\), make it the negative mirror image of \(0 < x < 3\). This gives: $$ f_s(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \ -1 & -1 < x < 0 \ 0 & -3 < x < -1 \end{cases} $$

Sketching the Periodic Function

Extend the given function with period \(b\). Repeat the function periodically every \(3\) units: $$ f_p(x) = f(x) \text{ where } f(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \end{cases} $$ The periodic extension is: $$ f_p(x+3) = f_p(x) $$

Fourier Series for the Even Function

The Fourier series of an even function only includes cosine terms. Compute the coefficients: $$ a_0 = \frac{1}{b} \int_{0}^{b} f_c(x) \, dx $$ $$ a_n = \frac{2}{b} \int_{0}^{b} f_c(x) \cos\left(\frac{n \pi x}{b}\right) \, dx $$ The Fourier series is: $$ f_c(x) \sim a_0 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n \pi x}{3}\right) $$

Fourier Series for the Odd Function

The Fourier series of an odd function only includes sine terms. Compute the coefficients: $$ b_n = \frac{2}{b} \int_{0}^{b} f_s(x) \sin\left(\frac{n \pi x}{b}\right) \, dx $$ The Fourier series is: $$ f_s(x) \sim \sum_{n=1}^{\infty} b_n \sin\left(\frac{n \pi x}{3}\right) $$

Fourier Series for the Periodic Function

For the function with period \(b\), both cosine and sine terms will be used. Compute the coefficients: $$ a_0 = \frac{1}{b} \int_{0}^{b} f_p(x) \, dx $$ $$ a_n = \frac{2}{b} \int_{0}^{b} f_p(x) \cos\left(\frac{2 n \pi x}{b}\right) \, dx $$ $$ b_n = \frac{2}{b} \int_{0}^{b} f_p(x) \sin\left(\frac{2 n \pi x}{b}\right) \, dx $$ The Fourier series is: $$ f_p(x) \sim a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{2 n \pi x}{3}\right) + b_n \sin\left(\frac{2 n \pi x}{3}\right) \right] $$

Key concepts

Even Function

An even function is symmetric about the y-axis. This means that for every point \(x\) on the function, there is a point \(-x\) that is equal in value. Mathematically, a function \(f(x)\) is even if it satisfies the condition \(f(x) = f(-x)\).
In the given exercise, we extended the function defined over \(0 < x < 3\) to create an even function over one period of \(-3 < x < 3\). This was done by reflecting the function across the y-axis, thereby creating a mirror image.

• Given function: \( f(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \end{cases}\)
• Extended even function over \(-3 < x < 3\): \( f_c(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \ 1 & -1 < x < 0 \ 0 & -3 < x < -1 \end{cases}\)

Odd Function

An odd function has rotational symmetry about the origin. For an odd function, the value at any point \(-x\) is the negative of the function value at \(x\). Mathematically, a function \(f(x)\) is odd if \(f(x) = -f(-x)\).
For the exercise, to extend the initial function to an odd function over one period \(-3 < x < 3\), we created the negative mirror image over \(-3 < x < 0\).

• Given function: \( f(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \end{cases}\)
• Extended odd function over \(-3 < x < 3\): \( f_s(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \ -1 & -1 < x < 0 \ 0 & -3 < x < -1 \end{cases}\)

Periodic Function

A periodic function repeats at regular intervals, or periods. Mathematically, a function \(f(x)\) is periodic with period \(T\) if \(f(x+T) = f(x)\) for all \(x\).
For the given exercise, the periodic function \( f_p(x) \) was created by repeating the original function every \(b\) units, where \(b = 3\). This means the function repeats itself every 3 units.

• Given function: \( f(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \end{cases}\)
• Periodic extension: \( f_p(x) = f(x) \text{ where } f(x) = \begin{cases} 1 & 0 < x < 1 \ 0 & 1 < x < 3 \end{cases} \)
• Repeated every 3 units: \( f_p(x+3) = f_p(x) \)

Fourier Coefficients

Fourier coefficients are essential in breaking down a periodic function into a sum of sines and cosines. There are different types of coefficients like \(a_0\), \(a_n\), and \(b_n\).
In general, they are computed as follows:

• For even functions:
  \(a_0 = \frac{1}{b} \int_{0}^{b} f_c(x) \, dx\)
  \(a_n = \frac{2}{b} \int_{0}^{b} f_c(x) \, cos\left(\frac{n \pi x}{b}\right) \, dx\)
• For odd functions:
  \(b_n = \frac{2}{b} \int_{0}^{b} f_s(x) \, sin\left(\frac{n \pi x}{b}\right) \, dx\)
• For general periodic functions:
  \(a_0 = \frac{1}{b} \int_{0}^{b} f_p(x) \, dx\)
  \(a_n = \frac{2}{b} \int_{0}^{b} f_p(x) \, cos\left(\frac{2 n \pi x}{b}\right) \, dx\)
  \(b_n = \frac{2}{b} \int_{0}^{b} f_p(x) \, sin\left(\frac{2 n \pi x}{b}\right) \, dx\)

Cosine Series

A cosine series describes a periodic even function solely using cosine terms. The cosine terms arise because even functions are symmetric and only need cosines to represent their symmetry.
The general form is:
\[ f_c(x) \sim a_0 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n \pi x}{b}\right) \]
The coefficients \(a_0\) and \(a_n\) are calculated using:
\[ a_0 = \frac{1}{b} \int_{0}^{b} f_c(x) \, dx \]
\[ a_n = \frac{2}{b} \int_{0}^{b} f_c(x) \, cos\left(\frac{n \pi x}{b}\right) \, dx \]
This method allows us to break down the even function into component cosines of different frequencies.

Sine Series

A sine series represents a periodic odd function using sine terms only. Since odd functions are inherently antisymmetric (i.e., they satisfy \( f(-x) = -f(x) \)), sines are ideal as they also exhibit this symmetry.
The general form is:
\[ f_s(x) \sim \sum_{n=1}^{\infty} b_n \sin\left(\frac{n \pi x}{b}\right) \]
The coefficients \(b_n\) are given by:
\[ b_n = \frac{2}{b} \int_{0}^{b} f_s(x) \, sin\left(\frac{n \pi x}{b}\right) \, dx \]
This simplifies the representation of odd functions in terms of their frequency components.