Question

Given \(f(x)=x\) for \(0

Short answer

Periodically extend \( f(x) = x \) to get \( f_c \) (even) and \( f_s \) (odd), then use Fourier series expansions with calculated coefficients.

Step-by-step solution

Understand the Functions

The goal is to sketch and expand both an even and an odd periodic function based on the given function: - Given function: \( f(x) = x \) for \( 0 < x < 1 \)

- Sketch the Even Function \( f_c \)

An even function is symmetric about the y-axis. 1. Reflect \( f(x) \) over the y-axis for \( -1 < x < 0 \), which gives: \( f_c(x) = -x \). 2. The resulting function between \( -1 < x < 1 \) is \( f_c(x) = -x \) for \( -1 < x < 0 \) and \( f_c(x) = x \) for \( 0 < x < 1 \). 3. Extend this function periodically with period 2.

- Write the Even Function in terms of Series

Since \( f_c \) is periodic with period 2, the Fourier cosine series for \( f_c \) is: \[ f_c(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n \pi x) \] Here's how to find \( a_0 \) and \( a_n \): \[ a_0 = \frac{1}{2} \int_{-1}^{1} f_c(x) dx = \frac{1}{2} \,0 \] \[ a_n = \int_{0}^{1} x \cos(n \pi x) dx \] Calculate it step by step.

- Sketch the Odd Function \( f_s \)

An odd function is symmetric about the origin. 1. Reflect \( f(x) \) over both the y-axis and the x-axis for \( -1 < x < 0 \), giving: \( f_s(x) = x \) for \( -1 < x < 0 \). 2. The resulting function between \( -1 < x < 1 \) is \( f_s(x) = -x \) for \( -1 < x < 0 \) and \( f_s(x) = x \) for \( 0 < x < 1 \). 3. Extend this function periodically with period 2.

- Write the Odd Function in terms of Series

Since \( f_s \) is periodic with period 2, the Fourier sine series for \( f_s \) is: \[ f_s(x) = \sum_{n=1}^{\infty} b_n \sin(n \pi x) \] Here's how to find \( b_n \): \[ b_n = \int_{0}^{1} x \sin(n \pi x) dx \] Calculate it step by step.

- Calculate Fourier Coefficients for the Cosine Series

Find \( a_n \) using integration: \[ a_n = \int_{0}^{1} x \cos(n \pi x) dx \] This integral can be solved using integration by parts.

- Calculate Fourier Coefficients for the Sine Series

Find \( b_n \) using integration: \[ b_n = \int_{0}^{1} x \sin(n \pi x) dx \] This integral can be solved using integration by parts.

- Write Down the Final Series

Substitute the computed \( a_n \) and \( b_n \) to express \( f_c \) and \( f_s \) as their respective Fourier series.

Key concepts

Fourier Cosine Series

The Fourier Cosine Series is used to represent an even function as an infinite sum of cosine functions. Given an even periodic function, the Fourier Cosine Series expansion utilizes only the cosine terms, effectively capturing the symmetry of even functions about the y-axis. The general form of the Fourier Cosine Series is:
\[ f_c(x) = a_0 + \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n \pi x) \]
Here,\( a_0 \) is the zeroth Fourier coefficient, which signifies the average value of the function over a period. The coefficients \(a_n \) can be calculated using:
\[ a_n = \int_{0}^{1} x \cos(n \pi x) dx \]
Integration by parts helps in evaluating these integrals. The resulting series can model any even function by capturing all the significant patterns and periodicity within the given range.

Fourier Sine Series

The Fourier Sine Series is key to describing an odd function using sine terms. Unlike cosine series, the sine series captures the symmetry of odd functions by reflecting the function about the origin. The general form of the Fourier Sine Series is:
\[ f_s(x) = \sum_{n=1}^{\tt} b_n \sin(n \pi x) \]
In this series, the coefficients \(b_n \) are crucial in defining the amplitude of each sine term. They are determined by:
\[ b_n = \int_{0}^{1} x \sin(n \pi x) dx \]
These integrals can also be computed using integration by parts. By summing the weighted sine terms, the series effectively models any odd function, ensuring each term aligns with the respective function's symmetry about the origin.

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals or periods. Mathematically, a function \( f(x) \) is periodic with period \( T \) if:
\[ f(x+T) = f(x) \] for all \( x \). These types of functions often arise in physical systems that exhibit repetitive behavior over time.
Examples of such functions include:

• Sine and cosine functions
• Square waves
• Triangular waves

Periodic functions are foundational in the study of Fourier series because they allow for the decomposition of complex, repetitive signals into simpler sine and cosine components. This decomposition is instrumental in numerous applications such as signal processing, vibration analysis, and acoustics.

Even and Odd Functions

Understanding even and odd functions is essential when working with Fourier series. Even functions exhibit symmetry about the y-axis, meaning their values mirror on either side of the axis. Mathematically, a function \( f(x) \) is even if:
\[ f(-x) = f(x) \] Examples include:

• Cosine function
• Parabolic function

Odd functions, on the other hand, have origin symmetry, meaning their values reflect through the origin with a sign change. Mathematically, a function \(f(x) \) is odd if:
\( f(-x) = -f(x) \) Examples include:

• Sine function
• Linear functions passing through the origin

Knowing these properties helps in correctly expanding functions using Fourier series, ensuring accurate representation through appropriate series. Even functions use Fourier Cosine Series while odd functions utilize Fourier Sine Series, respectively.