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)=x^{2},-\pi

Short answer

Find the Fourier coefficients for the function. Combine coefficients into a Fourier series. Repeat steps for both intervals.

Step-by-step solution

- Sketch Several Periods of the Given Functions

Start by graphing multiple periods of each given function to visualize their periodic nature: (a) For the function \(f(x) = x^{2}\) on the interval \(-\pi < x < \pi\), sketch \(y = x^2 \) from \(-\pi\) to \(\pi\) and then repeat this segment.(b) For the function \(f(x) = x^{2}\) on the interval \(0 < x < 2\pi\), sketch \(y = x^2 \) from \(0\) to \(2\pi\) and then repeat this segment.

- Write the General Form of the Fourier Series

Identify the general form of the Fourier Series for both sine-cosine and complex exponential forms. The sine-cosine form: \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \] The complex exponential form: \[ f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx} \]

- Calculate Fourier Coefficients for Part (a)

For \(f(x)=x^2\) over \(-\pi < x < \pi\):1. Calculate the Fourier coefficients \(a_0\), \(a_n\), and \(b_n\):\[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} x^2 \, dx \]\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) \, dx \]\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \sin(nx) \, dx \] 2. Perform the integrations and simplify. Note \(b_n\) terms will be zero due to the integrand's symmetry.3. Obtain the sine-cosine series and convert it to the complex exponential form.

- Calculate Fourier Coefficients for Part (b)

For \(f(x)=x^2\) over \(0 < x < 2\pi\):1. Calculate the Fourier coefficients \(a_0\), \(a_n\), and \(b_n\):\[ a_0 = \frac{1}{2\pi} \int_{0}^{2\pi} x^2 \, dx \]\[ a_n = \frac{1}{\pi} \int_{0}^{2\pi} x^2 \cos(nx) \, dx \]\[ b_n = \frac{1}{\pi} \int_{0}^{2\pi} x^2 \sin(nx) \, dx \] 2. Perform the integrations and simplify. Note \(b_n\) terms will be zero due to the integrand's symmetry.3. Obtain the sine-cosine series and convert it to the complex exponential form.

- Final Assembly of Fourier Series

Combine the calculated coefficients to form the complete Fourier series representation in both sine-cosine and complex exponential forms for both parts (a) and (b).

Key concepts

Fourier Coefficients

Fourier coefficients are the building blocks of a Fourier series, helping to break down complex periodic functions into simpler sine and cosine components.

To determine these coefficients, we use integration over one period of the function. The coefficients can be defined as follows:

• The coefficient \(a_0\) represents the average value (DC component).
  
• l\(a_n\) and \(b_n\) are the Fourier coefficients for the cosine and sine terms respectively, indicating the amplitude of the respective sine and cosine functions at harmonic frequencies nth times the base frequency.
  

These coefficients help in forming the sine-cosine and complex exponential forms of the Fourier series.

Sine-Cosine Form

The sine-cosine form of the Fourier series is an expansion of a function in terms of sine and cosine functions. It is represented as:

\[ f(x) = a_0 + \sum_{n=1}^{\infty} \( a_n \cos(nx) + b_n \sin(nx) \) \]
Where \(a_0\) is the mean value (DC component), \(a_n\) are the coefficients of the cosine terms, and \(b_n\) are the coefficients of the sine terms.

Steps to obtain this form are:

• Determine \(a_0\) using integration over one period:
  \[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \ dx \]
  
• Calculate \(a_n\):
  \[ a_n = \frac{1}{\bklash \pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \ dx \]
  


• Find \(b_n\):
  \[ b_n = \frac{1}{\bklash \pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \ dx \]
  

This form is advantageous as it allows breaking down a function into easily interpretable sinusoidal components.

Complex Exponential Form

The complex exponential form uses the Euler's formula to present a Fourier series with complex exponentials rather than sines and cosines. It is represented as:

\[ f(x) = \sum_{n=-\bkashefty}^{\bkashefty} c_n e^{inx} \]
Where \(c_n\) are the complex Fourier coefficients. This form is derived from the sine-cosine form using Euler's identities:

• \(e^{inx} = \cos(nx) + i \sin(nx)\)
  
• \(e^{-inx} = \cos(nx) - i \sin(nx)\)
  

The complex form can be expressed as:
\(c_n = \frac{1}{2\bkash ne\bkash\rightarrow}efty} \int_{-\bkash ne}^{\bkash ne} f(x) e^{-inx} \ dx\)

Benefits of this form include compactness and simplicity in mathematical manipulations such as differentiations and integrations.

Integration

Integration is crucial in finding the Fourier coefficients. The process involves:

• Definite integrals over one period of the function.
  
• Using symmetrical properties of functions to simplify integrals.
  

For example, to determine \(a_0\) for a function \(f(x)\) over the interval \( -\bkash\rightarrow\bkash\bkash efty} to \bkash \rightarrowefty} \):

• \[ a_0 = \bkash\bkash\bkash{1{2\bkash}}\bkash\braket} {}} \bkash f(x)\bkash x\]
  

Similar steps are used for \(a_n\) and \(b_n\). Integration allows us to weigh various harmonic components of a function accurately.