Question

(a) Sketch the graph of the given function for three periods. (b) Find the Fourier series for the given function. $$ f(x)=\left\\{\begin{array}{lr}{x+1,} & {-1 \leq x < 0,} \\ {1-x,} & {0 \leq x < 1 ;}\end{array} \quad f(x+2)=f(x)\right. $$

Short answer

Answer: The Fourier series representation of the given function is $$ f(x) = \frac{1}{4} + \sum_{n=1}^{\infty} \left[\frac{1}{n\pi}\left(\sin(n\pi)-n\pi-(-1)^{n+1}\right)\cos(\frac{n\pi x}{2}) + \frac{2}{n\pi}\left(\cos(\frac{n\pi}{2})-\cos(n\pi)+(-1)^n \right)\sin(\frac{n\pi x}{2})\right] $$

Step-by-step solution

Draw the graph for one period

First, let's sketch the graph for one period, i.e., from x = -1 to x = 1. For x in the interval [-1, 0), the function is equal to x + 1, which is a straight line with a positive slope. For x in the interval [0, 1), the function is equal to 1 - x, which is a straight line with a negative slope.

Draw the graph for three periods

To draw the graph for three periods, we need to replicate the graph for one period to the left and right of the original graph. So, we will draw a piecewise function with the same pattern for x in the intervals [-3, -2), and [1, 2).

Find the constant term a0

To find the constant term a0 in the Fourier series, we need to use the formula: $$ a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(x)dx $$ where T is the period of the function, which is 2 in our case. Now, we need to find the definite integral for both intervals of our function: $$ a_0 = \frac{1}{2} \left[ \int_{-1}^{0} (x+1)dx + \int_{0}^{1} (1-x)dx \right] $$

Calculate the constant a0

Now, let's integrate the functions and evaluate the results for definite integral: $$ a_0 = \frac{1}{2} \left[ \left[ \frac{1}{2}x^2 + x \right]_{-1}^0 + \left[ x - \frac{1}{2}x^2 \right]_0^1 \right] = \frac{1}{2} [(0 - \frac{1}{2}) + (1 - \frac{1}{2})] = \frac{1}{2} $$

Formula for cosine coefficients an

To find the Fourier coefficients an, we use the following formula: $$ a_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x)\cos(nx/T)dx $$ where T is the period of the function, which is 2. Again, we need to find the definite integral for both intervals of our function: $$ a_n = \frac{1}{2} \left[ \int_{-1}^{0} (x+1)\cos(\frac{n\pi x}{2})dx + \int_{0}^{1} (1-x)\cos(\frac{n\pi x}{2})dx \right] $$

Calculate the cosine coefficients an

Integrating the functions and evaluating the results for the definite integrals, we get: $$ a_n = \frac{1}{n\pi}\left[\sin(n\pi)-n\pi-(-1)^{n+1}\right] $$

Formula for sine coefficients bn

Similarly, to find the Fourier coefficients bn, we use: $$ b_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x)\sin(nx/T)dx $$ Again, we need to find the definite integral for both intervals of our function: $$ b_n = \frac{1}{2} \left[ \int_{-1}^{0} (x+1)\sin(\frac{n\pi x}{2})dx + \int_{0}^{1} (1-x)\sin(\frac{n\pi x}{2})dx \right] $$

Calculate the sine coefficients bn

Integrating the functions and evaluating the results for the definite integrals, we get: $$ b_n = \frac{2}{n\pi}\left[\cos(\frac{n\pi}{2})-\cos(n\pi)+(-1)^n \right] $$

Write the Fourier series for f(x)

Now, we have all the Fourier coefficients (a0, an, and bn). We can write the Fourier series for f(x) using these coefficients: $$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[a_n\cos(\frac{n\pi x}{T}) + b_n \sin(\frac{n\pi x}{T})\right] $$ Substituting the calculated coefficients (a0, an, and bn), we get the Fourier series representation of the given function: $$ f(x) = \frac{1}{4} + \sum_{n=1}^{\infty} \left[\frac{1}{n\pi}\left(\sin(n\pi)-n\pi-(-1)^{n+1}\right)\cos(\frac{n\pi x}{2}) + \frac{2}{n\pi}\left(\cos(\frac{n\pi}{2})-\cos(n\pi)+(-1)^n \right)\sin(\frac{n\pi x}{2})\right] $$

Key concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by different sub-functions within specific intervals. Imagine you have a line that changes its rules of formation depending on the section of the x-axis it is on. That's the essence of piecewise functions.

In the given exercise, the function described is a classic example: it behaves differently before and after the x=0 threshold. Understanding these functions is crucial, as they are a staple in mathematics, representing systems that behave differently under various conditions. To sketch them, you visualize each sub-function in its interval and connect these together, ensuring they meet at the specified points.

Skilled with graphing these functions? It’s important for understanding and analyzing real-life phenomena, such as transitions in physics or different tax brackets in economics.

Trigonometric Series

A trigonometric series is a series of waves—sines and cosines, to be precise—that can represent periodic functions through an infinite sum. Imagine an orchestra, where each instrument plays a note at a specific frequency to create a complex sound wave. This is similar to how a trigonometric series combines basic sine and cosine functions to reproduce more intricate periodic shapes.

When we talk about a Fourier series, we are essentially discussing a kind of trigonometric series that is specifically designed to approximate periodic functions, no matter how complex. The ability to break down a function into a trigonometric series is a powerful tool in mathematics, used in fields like signal processing and quantum mechanics.

Integral Calculus

Integral calculus is one of the foundational operations in calculus, alongside differentiation. If differentiation is about finding rates and slopes, integration is about accumulation and area. It lets us add up tiny pieces to determine the whole, or in technical terms, the area under a curve.

In our exercise, integral calculus is used to calculate the Fourier coefficients, which are essentially the weights of the various sine and cosine waves in our trigonometric series. To find the constant term and the coefficients, we integrate over the function's period, essentially summing up the influence of each sine and cosine component across the interval. This process links the piecewise function to its trigonometric series representation through a precise mathematical framework.

Fourier Coefficients

Fourier coefficients are the mathematical keys to unlocking the composition of a periodic function in terms of its sin and cos components. Think of them as the recipe for a complex dish, detailing the exact amounts of each ingredient needed to recreate it perfectly. In the Fourier series framework, these coefficients determine the amplitude and phase of each sine and cosine wave, essentially characterizing the periodic function's DNA.

Computing these values involves integrating the function multiplied by sine or cosine functions over one period. The answers, which are the Fourier coefficients, tell you how much of each 'flavor' (sine or cosine of a particular frequency) you'll need to mix together to perfectly match the original function. Mastering Fourier coefficients is learning the language of signals and systems, granting insight into a vast array of physical and abstract phenomena.