Question

Each of the functions is given over one period. For each function, sketch several periods and decide whether it is even or odd. Then use (9.4) or (9.5) to expand it in an appropriate Fourier series. $$ f(x)=\left\\{\begin{array}{ll} 1, & -1

Short answer

Under consruction

Step-by-step solution

- Understand the Given Function

The function is defined as follows:

Key concepts

Even and Odd Functions

Before diving into sketching and Fourier series, it's vital to know if the function is even or odd. This will simplify calculations. An even function is symmetric about the y-axis. Mathematically, a function is even if: \[ f(x) = f(-x) \ \text{for all } x \text{ in the domain.} \ \]In contrast, an odd function is symmetric about the origin. Mathematically, it's defined as: \[ f(x) = -f(-x) \ \text{for all } x \text{ in the domain.} \ \]For the given function, you need to test symmetry by evaluating \[ f(x) \ \text{and} \ f(-x). \ \]Here's how to do it: 1. If \( f(x) = 1 \) for \( -1 < x < 1 \), then check the same interval with \( x \) replaced by \( -x \). This means seeing what happens to \( f(-x) \). 2. Similarly, apply this for the given piece: \( f(x) = 0 \) for \( -2 < x < -1 \) and \( 1 < x < 2 \).For this exercise, if you find \( f(x) eq f(-x) \) but \( f(x) = -f(-x) \), the function is odd. If these don't hold, further analysis may be required to classify it. Knowing this helps to use the right formula for Fourier series.

Function Sketching

Sketching helps visualize the function over multiple periods. Here's a simple approach:

• Plot the given piecewise parts on graph paper or using a software tool.
• For the interval \( -2 < x < 2 \), draw the function values according to the problem.
• Since the function's defined on \( -2 < x < 2 \), repeat it for several periods. Just copy and paste data points across left and right intervals.

This repetitive pattern gives a clear visual. For instance, you repeat the function from \( -4 < x < -2 \) and \( 2 < x < 4 \). Function sketching isn't just helpful for seeing the pattern; it clarifies if you correctly identified the function as even or odd. Sketching also makes Fourier series easier to apply and understand.

Fourier Series

Expanding a function into a Fourier series involves expressing it as a sum of sine and cosine functions. It’s essential to use the right series based on if the function is even or odd:

• Even Function: Use only cosine terms.
• Odd Function: Use only sine terms.

For our function, after classifying it (either even or odd), we choose the appropriate form:For an even function: \[ f(x) = a_0 + \ \text{sum of} \ a_n \ \text{cosine terms,} \ \]where coefficients are calculated by: \[ a_0 = \frac{1}{\text{period}} \ \text{integral over one period}\f(x) dx \]\[ a_n = \frac{2}{\text{period}} \ \text{integral over one period}\f(x) \ \text{cos}(nx) dx \]For an odd function: \[ f(x) = \ \text{sum of} \ b_n \ \text{sine terms,} \ \]where coefficients are calculated by: \[ b_n = \frac{2}{\text{period}} \ \text{integral over one period}\f(x) \ \text{sin}(nx) dx \]Lastly, integrate over the given intervals to find these coefficients. Once obtained, sum the series to represent your function accurately.