Question

Parts (a) and \((b)\), you are given in each case one period of a function. Sketch several periods of the function and expand it in a sine-cosine Fourier series, and in a complex exponential Fourier series. (a) \(f(x)=2-x,-2

Short answer

Fourier series coefficients for both parts depend on the given integral forms. Use sine-cosine coefficients to form the complex exponential series.

Step-by-step solution

- Identify the Period

First, determine the period of the function for each case. For part (a), the period is from -2 to 2, and for part (b), the period is from 0 to 4.

- Sketch Several Periods

Plot the function over several periods for better visualization. For both cases, repeat the function over multiple cycles based on their respective periods.

- Fourier Series Coefficients for Sine-Cosine Series

For the sine-cosine Fourier series, we need to find the coefficients. Given the function is defined over one period, calculate the coefficients using the integrals:\( a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(x) \, dx \)\( a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \cos\left(\frac{2\pi nx}{T}\right) \, dx \)\( b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \sin\left(\frac{2\pi nx}{T}\right) \, dx \)

- Compute the Coefficients for Part (a)

For part (a), the period is 4. Calculate\(T = 4\)\( a_0 = \frac{1}{4} \int_{-2}^{2} (2 - x) \, dx = 0\)\( a_n = \frac{2}{4} \int_{-2}^{2} (2 - x) \cos\left(\frac{n\pi x}{2} \right) \, dx = \frac{8}{n^2 \pi^2}(\cos(n\pi)-1)\)\( b_n = \frac{2}{4} \int_{-2}^{2} (2 - x) \sin\left(\frac{n\pi x}{2} \right) \, dx = \frac{8(-1)^n}{n\pi}\)

- Compute the Coefficients for Part (b)

For part (b), the period is 4. Calculate\(T = 4\)\( a_0 = \frac{1}{4} \int_{0}^{4} (2 - x) \, dx = 0\)\( a_n = \frac{2}{4} \int_{0}^{4} (2 - x) \cos\left(\frac{n\pi x}{2} \right) \, dx = \frac{8}{n^2\pi^2}(\cos(2n\pi)-1)\)\( b_n = \frac{2}{4} \int_{0}^{4} (2 - x) \sin\left(\frac{n\pi x}{2} \right) \, dx = \frac{8(-1)^n}{n\pi}\)

- Fourier Series Representation

Using the coefficients, write the Fourier series representations:For part (a):\( f(x) = \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{2}\right) + b_n \sin\left(\frac{n\pi x}{2}\right) \)For part (b):\( f(x) = \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{2}\right) + b_n \sin\left(\frac{n\pi x}{2} \right) \)

- Complex Exponential Fourier Series

To find the complex exponential Fourier series, use the coefficients from the previous step and the formula:\( f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n \frac{2\pi x}{T}} \)where \( c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x) e^{-i n \frac{2\pi x}{T}} \, dx \)

- Calculate Complex Coefficients

For both parts, the complex Fourier coefficients are the same as the calculated sine-cosine coefficients transformed into the exponential form, thus:\( c_n = \frac{a_n - i b_n}{2} \)

- Final Representations

Write the final complex exponential Fourier series for both parts:For part (a):\( f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n \frac{\pi x}{2}} \)For part (b):\( f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n \frac{\pi x}{2}} \)

Key concepts

Sine-Cosine Fourier series

The sine-cosine Fourier series is a way to represent a periodic function using a combination of sine and cosine terms. This method expands the given function into an infinite series of these trigonometric functions, which makes it easier to analyze and approximate.

The general form of a sine-cosine Fourier series for a function over a period \( T \) is given by:
\[ f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(\frac{2\pi nx}{T}) + b_n \sin(\frac{2\pi nx}{T})) \]

Here’s what each term represents:

• \( a_0 \): The average value (DC component) of the function over one period.
• \( a_n \): The coefficients of the cosine terms, representing the even harmonics.
• \( b_n \): The coefficients of the sine terms, representing the odd harmonics.

To find these coefficients, we use the following formulas:
\[ a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(x) \, dx \]
\[ a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \cos(\frac{2\pi nx}{T}) \, dx \]
\[ b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \sin(\frac{2\pi nx}{T}) \, dx \]
These integrals calculate the amount of each sine and cosine component present in the original function.

Complex Exponential Fourier series

The complex exponential Fourier series is an alternative to the sine-cosine series that uses complex exponentials, which often simplifies the math. The basic idea is to express a periodic function as a sum of complex exponentials:
\[ f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n \frac{2\pi x}{T}} \]

In this formula:

• \( c_n \): The complex Fourier coefficients, which provide the amplitude and phase information for each exponential component.
• \( e^{i n \frac{2\pi x}{T}} \): Complex exponentials, representing the harmonic frequencies of the original function.

To find the complex Fourier coefficients \( c_n \), we use the formula:
\[ c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x) e^{-i n \frac{2\pi x}{T}} \, dx \]

If you have already found the Fourier coefficients \( a_n \) and \( b_n \) from the sine-cosine series, you can convert them to the complex form using:
\[ c_n = \frac{a_n - i b_n}{2} \]
This transformation captures the same harmonic content in a more compact and often more convenient form.

Fourier coefficients

Fourier coefficients are values that describe the weights of the sine, cosine, and exponential terms in a Fourier series. These coefficients are essential to reconstructing the original function from its Fourier series representation.

For the sine-cosine Fourier series, the coefficients are:

• \( a_0 \): The average value (DC component) of the function over one period.
• \( a_n \): The Fourier cosine coefficients, which can be calculated using \( a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \cos(\frac{2\pi nx}{T}) \, dx \).
• \( b_n \): The Fourier sine coefficients, which can be calculated using \( b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \sin(\frac{2\pi nx}{T}) \, dx \).

For the complex exponential Fourier series, the coefficients are expressed as:
\[ c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x) e^{-i n \frac{2\pi x}{T}} \, dx \]
These coefficients encode both amplitude and phase information, offering a more nuanced view of the function's frequency content. By understanding these coefficients, you can gain deeper insights into the behavior and characteristics of periodic functions.