# Helpful Revision for Fourier Series

### On this page

- Properties of Sine and Cosine Graphs
- Periodic Functions
- Continuity
- Split Functions
- Summation Notation
- Useful Integrals

### Background

From previous chapters:

Sine and cosine curves

Even and odd functions

Integral of Sine and Cosine

This page contains some background information that will help you to better understand this chapter on Fourier Series.

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`.

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.

### Examples of Periodic Functions

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

*p*= 2π

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

### Useful Background

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

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

*p*= 2

Graph of saw-tooth waveform with period `2`

### Useful background

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

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}`

Answer

Graph of `f(t)`.

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

Answer

Graph of `f(t)`, a sawtooth wave.

### Useful Background

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}`

Answer

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}`

Answer

Graph of `f(t)`.

**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)`

Answer

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

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

Answer

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

**Notice** that the expression (2*n* − 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`

Answer

`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`

Answer

`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`