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# Helpful Revision for Fourier Series

### Background

From previous chapters:

You have seen most of this before, but I have included it here to give you some help before getting into the heavy stuff.

## Properties of Sine and Cosine Functions

These properties can simplify the integrations that we will perform later in this chapter.

### The Cosine Function

The function f(x) = cos x is an even function. That is, it is symmetrical about the vertical axis.

We have: cos(-x) = cos(x), and it follows that

int_(-pi)^picos theta\ d theta=0

We can see the integral must be 0 if we consider the curve of y = cos(x) for x=-pi to x=pi.

The integral of y=cos(x) for -pi <= x <= pi

The yellow "negative" portions of the graph when added to the green "positive" portion cancel each other out. They add to zero.

### The Sine Function

The function f(x) = sin x is an odd function. That is, it is symmetrical about the origin.

We have: sin(-x) = -sin(x), and

int_(-pi)^pi sin theta\ d theta=0

Once again, a consideration of the areas under the graph illustrates why it is so.

The integral of y=sin(x) for -pi <= x <= pi

Once again, the yellow negative portion is the same size as the green positive portion, so the sum is 0.

## Multiples of π for Sine and Cosine Curves

Consider the function y = sin x.

### Revision

For some background:

Sine and cosine curves

The curve y=sin(x) for 0 <= x <= 8pi

From the graph (or using our calculator), we can observe that:

sin(nπ) = 0,\  for n = 0, 1, 2, 3, ... (in fact, all integers)

sin{:((2n-1)pi)/2:}=(-1)^(n+1),\  for n = 0, 1, 2, 3, ... (in fact, all integers)

### Cosine case

Next, we consider the curve y = cos x

The curve y=cos(x) for 0 <= x <= 8pi

cos(2nπ) = 1,\  for n = 0, 1, 2, 3, ... (in fact, all integers)

cos[(2n − 1)π] = −1,\  for n = 0, 1, 2, 3, ... (in fact, all integers)

cos(nπ) = (−1)^n,\  for n = 0, 1, 2, 3, ... (in fact, all integers)

## Periodic Functions

A function f(t) is said to be periodic with period p if

f(t + p) = f(t)

for all values of t and if p > 0.

The period of the function f(t) is the interval between two successive repetitions.

### 1a. f(t) = sin t

p = 2π

The curve y=sin(x), which has period = 2pi

### Useful Background

Sine and cosine curves

For f(t) = sin t, we have: f(t) = f(t + 2π). The period is 2π.

### 1b. Saw tooth waveform, period = 2

p = 2

Graph of saw-tooth waveform with period 2

### Useful background

Straight lines

For this function, we have:

f(t) = 3t (for −1 ≤ t < 1)

f(t) = f(t + 2) [This expression indicates it is periodic with period 2.]

### 1c. Parabolic, period = 2

p = 2

Periodic graph based on parabola, with period 2.

### Useful background

Parabolas

For this function, we have:

f(t) = t^2 (for 0 ≤ t < 2)

f(t) = f(t + 2) [Indicating it is periodic with period 2.]

### 1d. Square wave, period = 4

p = 4

Graph of periodic square wave function.

For this function, we have:

f(t) = -3 for -1 ≤ t < 1 and 3 for 1 ≤ t < 3

f(t) = f(t + 4) [The period is 4.]

NOTE: In this example, the period p = 4. We can write this as 2L = 4.

In the diagram we are thinking of one cycle starting at  −2 and finishing at 2. For convenience when integrating later, we let L = 2 and the cycle goes from -L to L.

## Continuity

If a graph of a function has no sudden jumps or breaks, it is called a continuous function.

Examples:

### Useful Background

• sine functions
• cosine functions
• exponential functions
• parabolic functions

Finite discontinuity - a function makes a finite jump at some point or points in the interval.

Examples:

• Square wave function
• Saw tooth functions

## Split Functions

Much of the behaviour of current, charge and voltage in an AC circuit can be described using split functions.

### Examples of Split Functions

Sketch the following functions:

### Useful Background

2a. f(t)=\begin{cases} -t & \text{if} & -\pi \le t < 0 \\ t & \text{if} & 0 \le t < \pi\end{cases}

2b. f(t)=\begin{cases} t & \text{if} & 0 \le t < \pi \\ t-\pi & \text{if} & \pi \le t < 2\pi\end{cases}

Graph of f(t), a sawtooth wave.

### Useful Background

Parabolas

2c. f(t)=\begin{cases} (t+\pi)^2 & \text{if} & -\pi \le t < 0 \\ (t-\pi)^2 & \text{if} & 0 \le t < \pi\end{cases}

We note that f(0) = pi^2 = 9.8696....

Graph of f(t), an even function.

2d. f(t)=\begin{cases} t+\pi & \text{if} & -\pi \le t < -\frac{\pi}{2} \\ -1 & \text{if} & -\frac{\pi}{2} \le t < \frac{\pi}{2} \\ -t+\pi & \text{if} & \frac{\pi}{2} \le t < \pi\end{cases}

## Summation Notation

It is important to understand summation notation when dealing with Fourier series.

### Examples

Expand the following and simplify where possible:

3a. sum_(n=1)^3n/(n+1)

sum_(n=1)^3n/(n+1)= 1/2+2/3+3/4 =23/12

3b. sum_(n=1)^5(2n-1)

sum_{n=1}^5(2n-1) =1+3+5+7+9 =25

Notice that the expression (2n − 1) generates odd numbers. as n takes values 1, 2, 3, 4, 5.

If we want to generate even numbers, we would use 2n, as follows:

sum_{n=1}^5{2n} =2+4+6+8+10 =30

To generate alternate positive and negative numbers, we multiply the expression in the summation by (−1)n+1. For example:

sum_{n=1}^5(-1)^{n+1}(n) =1-2+3-4+5 =3

And here's a final example, giving us alternate positive and negative fractional terms with even denominators:

sum_{n=1}^5\frac{(-1)^{n+1}}{2n} =\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10} =\frac{47}{120}

3c. sum_(n=1)^5n^2a_n

sum_{n=1}^5 n^2 a_n =a_1+4a_2+9a_3+16a_4+25a_5

3d. sum_(n=1)^4(npit)/L

sum_(n=1)^4(npit)/L=(pit)/L+(2pit)/L+(3pit)/L+(4pit)/L

=(pit)/L(1+2+3+4)

=10(pit)/L

## Some Useful Integrals

The next 2 integrals are obtained from integration by parts and can be found in the Table of Common Integrals. We use them quite a bit in this Fourier Series chapter.

int t sin nt dt =1/n^2(sin nt-nt cos nt)+K

int t cos nt dt =1/n^2(cos nt+nt sin nt)+K