Question

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

Short answer

Defined even, odd, and periodic functions over period. Calculated Fourier series for each.

Step-by-step solution

Define the Even Function

The even function, denoted as \( f_{c}(x) \), will be symmetric around the y-axis. This means \( f_{c}(x) = f_{c}(-x) \). On the interval \(0 < x < b\), \( f_{c}(x) = f(x) \). For \( -b < x < 0 \), \( f_{c}(x) = f(-x) \).

Define the Odd Function

The odd function, denoted as \( f_{s}(x) \), will be anti-symmetric around the y-axis. This means \( f_{s}(x) = -f_{s}(-x) \). On the interval \(0 < x < b\), \( f_{s}(x) = f(x) \). For \( -b < x < 0 \), \( f_{s}(x) = -f(-x) \).

Define the Periodic Function

The periodic function, denoted as \( f_{p}(x) \), will repeat every \( b \) intervals. This means \( f_{p}(x+b) = f_{p}(x) \). On the interval \(0 < x < b\), \( f_{p}(x) = f(x) \). For any other interval, \( f_{p}(x) \) will repeat the given function.

Fourier Series for the Even Function

To find the Fourier series of the even function \( f_{c}(x) \), calculate the coefficients as: a_0 = \frac{1}{b} \int_{-b}^{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 becomes:\(f_c(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n \pi x}{b}\right).\)

Fourier Series for the Odd Function

To find the Fourier series of the odd function \( f_{s}(x) \), calculate the coefficients as: \(b_n = \frac{2}{b} \int_{0}^{b} f_{s}(x) \sin\left(\frac{n \pi x}{b}\right) \, dx\). The Fourier series becomes:\(f_s(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n \pi x}{b}\right).\)

Fourier Series for the Periodic Function

To find the Fourier series of the periodic function \( f_{p}(x) \), calculate the coefficients as: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{2n \pi x}{b}\right) \, dx,\(b_n = \frac{2}{b} \int_{0}^{b} f_{p}(x) \sin\left(\frac{2n \pi x}{b}\right) \, dx\). The Fourier series becomes:\(f_p(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{2n \pi x}{b} \right) + b_n \sin\left(\frac{2n \pi x}{b} \right)\right).\)

Key concepts

Even and Odd Functions

In mathematics, even and odd functions are types of functions that exhibit specific symmetrical properties. Understanding these functions is crucial for comprehending Fourier series.

• Even Functions: These functions are symmetric around the y-axis. The defining property is \( f_{c}(x) = f_{c}(-x) \). This means the function remains unchanged if you reflect it over the y-axis.
• Odd Functions: These functions are symmetric about the origin. They satisfy \( f_{s}(x) = -f_{s}(-x) \), implying that rotating them 180 degrees around the origin yields the same function.

Understanding these properties allows us to visualize their shapes and behavior better. For example, the even function for given \( f(x) \) in the interval \( 0 < x < b \) would mirror itself on the negative x-axis, while the odd function would invert symmetry on the negative side.
For example, the even extension of function \( f(x) \) over \( 0 < x < b \) would be mirrored in the range \( -b < x < 0 \) while the odd extension would be the negative of \( f(x) \) in this range.

Periodic Functions

Periodic functions repeat their values in regular intervals. They are fundamental in studying Fourier series because they help break down complex signals into simpler components.

• Definition: A function \( f_{p}(x) \) is periodic with period \( b \) if \( f_{p}(x + b) = f_{p}(x) \) for all x. This means the function repeats every interval of length \( b \).
• Application: To create a periodic function from \( f(x) \) defined on \( 0 < x < b \), we extend \( f(x) \) so that \( f_{p}(x) \) for the interval \( n*b < x < (n+1)*b \) exactly replicates \( f(x) \).

For example, the function given is defined on \( 0 < x < 1 \). Extending this to a periodic function with period \( b = 1 \), we get repetitions of this function over intervals like \( 1 < x < 2 \) and \( -1 < x < 0 \). Periodic functions help simplify the analysis by turning potentially complex behaviors into a series of repeating patterns.

Coefficient Calculation

Calculating Fourier coefficients is essential to constructing the Fourier series representation of a function. There are different formulas based on whether the function is even, odd, or neither.
For the Fourier series of an even function, coefficients are calculated as:
\[ a_0 = \frac{1}{b} \int_{-b}^{b} f_{c}(x) \, dx, \ a_n = \frac{2}{b} \int_{0}^{b} f_{c}(x) \cos\left(\frac{n \pi x}{b}\right) \, dx. \ \ \ \ \ \ \]
The series is then:
\[ f_c(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n \pi x}{b}\right). \ \ \]
For the Fourier series of an odd function, the coefficients are:
\[ b_n = \frac{2}{b} \int_{0}^{b} f_{s}(x) \ \sin\left(\frac{n \pi x}{b}\right) \, dx, \ \ \ \]
and the series is:
\[ f_s(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n \pi x}{b}\right). \ \ \]
For the periodic function, the coefficients for the Fourier series include both cosine and sine terms:
\[ a_0 = \frac{1}{b} \int_{0}^{b} f_{p}(x) \, dx, \ a_n = \frac{2}{b} \int_{0}^{b} f_{p}(x) \cos\left(\frac{2n \pi x}{b}\right) \, dx, \ b_n = \frac{2}{b} \int_{0}^{b} f_{p}(x) \sin\left(\frac{2n \pi x}{b}\right) \, dx. \ \ \ \]
The series looks like:
\[ f_p(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{2n \pi x}{b} \right) + b_n \sin\left(\frac{2n \pi x}{b} \right)\right) \ \ \ \]