Problem 8 - Calculus

Question

Find the \(n^{\text {th }}\). Fourier polynomial for the given functions, assuming them to be periodic with period \(2 \pi\) Graph the first three approximations with the original function. $$f(x)=x^{2}, \quad-\pi

Step-by-Step Solution

Answer

The Fourier series is \(f(x) = \frac{\pi^2}{3} + \sum_{n=1}^{\infty} \frac{-2(-1)^n}{n^2} \sin(nx) \).

Step 1: Define Fourier Series

The Fourier series of a function with period \(2\pi\) is given by:\[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \]where:\[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx \]\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \]\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \]

Step 2: Calculate the constant term \(a_0\)

Substitute \(f(x) = x^2\) into the integral for \(a_0\):\[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} x^2 \, dx \]This integral calculates to:\[ a_0 = \frac{1}{2\pi} \left[ \frac{x^3}{3} \right]_{-\pi}^{\pi} = \frac{1}{2\pi} \left( \frac{\pi^3}{3} - \left( -\frac{\pi^3}{3} \right) \right) = \frac{\pi^2}{3} \]

Step 3: Calculate coefficients \(a_n\)

Substitute \( f(x) = x^2 \) into the integral for \( a_n \):\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) \, dx \]Use integration by parts or symmetry to evaluate the integral, considering odd functions evaluate to zero over symmetric limits. Thus, \( a_n = 0 \) for all \( n \geq 1 \).

Step 4: Calculate coefficients \(b_n\)

Substitute \( f(x) = x^2 \) into the integral for \( b_n \):\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \sin(nx) \, dx \]Evaluate the integral using integration by parts. As a result, you find:\[ b_n = \frac{-2(-1)^n}{n^2} \]

Step 5: Write the Fourier Series

Compile the calculated coefficients into the Fourier series:\[ f(x) = \frac{\pi^2}{3} + \sum_{n=1}^{\infty} \frac{-2(-1)^n}{n^2} \sin(nx) \]

Step 6: Form First Three Fourier Polynomials

For the first three terms (\(n=1, 2, 3\)), the approximations are:- 1st approximation (\(n=1\)): \[ f_1(x) = \frac{\pi^2}{3} - 2\sin(x) \]- 2nd approximation (\(n=2\)): \[ f_2(x) = \frac{\pi^2}{3} - 2\sin(x) + \frac{1}{2}\sin(2x) \]- 3rd approximation (\(n=3\)): \[ f_3(x) = \frac{\pi^2}{3} - 2\sin(x) + \frac{1}{2}\sin(2x) - \frac{2}{9}\sin(3x) \]

Step 7: Plot and Compare Graphs

Now plot the original function \(f(x) = x^2\) and each of the first three Fourier approximations. Observe how each additional term ( = 1, 2, 3) gradually makes the approximation converge better towards the original function.

Key Concepts

Fourier coefficients

In the Fourier series, Fourier coefficients are crucial as they represent the weights of respective terms in the series. For a given function, these coefficients tell us how much of each harmonic (sine and cosine) frequency is present in the signal. The Fourier series for a function with period \(2\pi\) is written as: \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \] Here, the first coefficient, \(a_0\), is the average or mean value of the function over one period. It is calculated using the integral: \[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx \] For coefficients \(a_n\) and \(b_n\), they quantify the amplitude of the cosine and sine components, respectively, for each frequency \(n\). They are determined by:

• \( a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \)
• \( b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \)

These integrals may require techniques like symmetry or integration by parts to compute, especially for complicated functions.

Periodic functions

Periodic functions are a type of function that repeat their values at regular intervals or periods. In our exercise, the given function \( f(x) = x^2\) is assumed to be periodic with a period of \(2\pi\). This means that the function's behavior over the interval \(-\pi < x \leq \pi\) is mirrored for every subsequent interval of length \(2\pi\). The periodic nature of a function is essential for using Fourier series effectively. Because Fourier series aim to represent a function as a sum of sines and cosines, which are inherently periodic, it is crucial that the function itself possesses periodic characteristics.

• Period \( T \): The smallest positive interval over which the function repeats (here, \( T = 2\pi \)).
• Harmonics: These are multiples of the fundamental frequency (\(1/T\)), responsible for the oscillating nature of Fourier series.

Analyzing a function as periodic enables us to employ mathematical techniques that simplify complex wave-like patterns into basic harmonic components.

Integration by parts

Integration by parts is a technique used to integrate the product of two functions. This method is handy in calculating Fourier coefficients, especially the \(b_n\) terms of a Fourier series that involve sine functions. The formula for integration by parts is derived from the product rule for derivatives and is given by: \[ \int u \, dv = uv - \int v \, du \] Where \(u\) and \(dv\) are parts of the integrand \(u \, dv = f(x) \sin(nx) \, dx \).
When working with Fourier coefficients, strategic choice of \(u\) and \(dv\) can simplify the integral significantly:

• Choose \(u\) to be the polynomial part --- typically \(x^2\) in our problem.
• Choose \(dv\) to be the trigonometric function, \(\sin(nx) \, dx\).

The technique effectively breaks down the original problem into simpler parts, facilitating the computation of otherwise difficult integrals. In our context, integration by parts finally reveals patterns, such as resulting coefficients being zero or following predictable sequences due to periodic properties.