Problem 9 - 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. $$h(x)=\left\\{\begin{array}{ll}0 & -\pi

Step-by-Step Solution

Answer

The nth Fourier polynomial approximation is \( F_n(x) = \frac{\pi}{4} + \sum_{n=1}^{n} \frac{2(-1)^{n+1}}{n} \sin(nx) \).

Step 1: Define the Problem

We need to find the nth Fourier polynomial approximation of the piecewise function \( h(x) \) over the interval \( [-\pi, \pi] \) where the function is defined as \( 0 \) for \( -\pi < x \leq 0 \) and \( x \) for \( 0 < x \leq \pi \). Thereafter, plot the first three Fourier polynomial approximations and compare them to the original function.

Step 2: Determine Fourier Coefficients

For the piecewise function where the period is \( 2\pi \), calculate constants: - The average term \( a_0 \) using \( a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} h(x)\, dx \).- Cosine coefficients \( a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} h(x)\cos(nx)\,dx \). - Sine coefficients \( b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} h(x)\sin(nx)\,dx \).

Step 3: Calculate \( a_0 \)

Divide the integral: \( a_0 = \frac{1}{\pi} \left( \int_{-\pi}^{0} 0\, dx + \int_{0}^{\pi} x \, dx \right) \). Evaluate: \[ a_0 = \frac{1}{\pi} \left( 0 + \left[ \frac{x^2}{2} \right]_{0}^{\pi} \right) = \frac{\pi^2}{2\pi} = \frac{\pi}{2} \].

Step 4: Evaluate \( a_n \)

Evaluate the integral for \( a_n \): \[ a_n = \frac{1}{\pi} \left( \int_{0}^{\pi} x \cos(nx)\,dx \right) \]. Use integration by parts with \( u = x \) and \( dv = \cos(nx) dx \) to find \[ a_n = \frac{1}{\pi} \left( \frac{x \sin(nx)}{n}\bigg|_0^{\pi} - \int_{0}^{\pi} \frac{\sin(nx)}{n} dx \right) = 0 \]. Thus, \( a_n = 0 \) for all \( n \).

Step 5: Calculate \( b_n \)

Evaluate the integral for \( b_n \): \[ b_n = \frac{1}{\pi} \left( \int_{0}^{\pi} x \sin(nx)\,dx \right) \]. Use integration by parts with \( u = x \) and \( dv = \sin(nx) dx \) to find: \[ b_n = \left( -\frac{x \cos(nx)}{n} \bigg|_0^{\pi} + \int_{0}^{\pi} \frac{\cos(nx)}{n} dx \right) = \frac{2(-1)^{n+1}}{n} \].

Step 6: Construct the nth Fourier Polynomial

Using the Fourier coefficients, the nth Fourier polynomial is given by:\[ F_n(x) = \frac{a_0}{2} + \sum_{n=1}^{n} (a_n \cos(nx) + b_n \sin(nx)) \]. Substitute the calculated coefficients:\[ F_n(x) = \frac{\pi}{4} + \sum_{n=1}^{n} \frac{2(-1)^{n+1}}{n} \sin(nx) \].

Step 7: Graph the First Three Approximations

Plot the original function and the first three Fourier polynomial approximations (\( n=1, 2, 3 \)) using the expression \[ F_n(x) = \frac{\pi}{4} + \sum_{n=1}^{n} \frac{2(-1)^{n+1}}{n} \sin(nx) \]. Observe how the Fourier series approximates the piecewise function as \( n \) increases.

Key Concepts

Fourier Coefficients

Fourier coefficients are critical in approximating periodic functions using Fourier polynomials. These coefficients break down a complex function into sine and cosine terms. They are essentially the building blocks of Fourier series.
To find these coefficients, we use integrals. The process involves splitting the function into three parts: the average term, cosine coefficients, and sine coefficients.

• The average term, denoted by \(a_0\), is found by integrating the function over one period and dividing by the period length.
• The cosine coefficients, \(a_n\), involve integrating the product of the function and \(\cos(nx)\).
• The sine coefficients, \(b_n\), are derived from integrating the product of the function and \(\sin(nx)\).

Calculating these coefficients accurately is essential to construct an approximate representation of the original function in the form of a Fourier series. This series can then be used to model the function's behavior over its period.

Piecewise Function

A piecewise function is one that is defined by different expressions based on the input value. In our problem, the function \( h(x) \) is piecewise, with different expressions on the intervals \(-\pi < x \leq 0\) and \(0 < x \leq \pi\).
Piecewise functions are common in mathematics whenever a rule needs to change over different parts of a domain. They might look complex at first, but they allow for flexible modeling of real-world phenomena.

• In the negative interval, the function value is constant, \( h(x) = 0 \), which simplifies calculations.
• In the positive interval, the function is linear, \( h(x) = x \), allowing us to use basic algebraic techniques in computations.

Dealing with piecewise functions requires evaluating each segment separately, using the appropriate function expression for each interval.

Periodic Function

Periodic functions repeat their values in regular intervals or periods. In this exercise, the function \( h(x) \) is assumed to be periodic with the period \( 2\pi \).
Understanding periodicity is key to solving Fourier analysis problems, as this property ensures that the function can be modeled as a sum of sine and cosine functions which themselves are periodic.
Periodic functions like sine and cosine have the advantage of being predictable over a set interval, making them ideal to model repeating phenomena such as sound waves or vibrations.

• The period for a repeating cycle of the function is \( 2\pi \), meaning it completes one full cycle over this interval.
• Knowledge of periodicity helps us understand how to extend functions beyond their initial interval.

In Fourier analysis, identifying and using the period gives a clear boundary for calculating the coefficients and constructing the Fourier series.

Integration by Parts

Integration by parts is a technique used in calculus to solve integrals involving products of functions. This concept is central when dealing with Fourier coefficients, specifically when finding \(a_n\) and \(b_n\).
The rule can be described as: \(\int u \, dv = uv - \int v \, du\), where \(u\) and \(dv\) are chosen parts of the original integrand.

• Applying integration by parts requires choosing \(u\) and \(dv\) carefully to simplify the resulting integral.
• For the Fourier coefficients, typical choices are \(u = x\) and either \(dv = \cos(nx)dx\) or \(dv = \sin(nx)dx\).

Using integration by parts in Fourier analysis allows us to handle otherwise challenging integrations by breaking them into simpler pieces. It is a versatile tool that eases the process of finding accurate coefficients for approximating periodic functions.