# Helpful Revision for 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.

### Background

From previous chapters:

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

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

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, 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

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)

Next, we consider the curve y = cos x

 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.

### 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:

### 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.

### 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.

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)={ {: (-t,if -pi<=t < 0),(t,if 0 <=t < pi) :}

2b. f(t)={ {:(t,if 0 <=t < pi), (t-pi, if pi <= t <2pi):}

### Useful Background

Parabolas

2c. f(t)={ {: ((t+pi)^2,if -pi <=t <0),((t-pi)^2,if 0 <= t < pi) :}

2d. f(t)={(t+pi, if -pi <= t < -pi/2),(-1, if -pi/2 <= t < pi/2),(-t+pi, if pi/2 <= t < pi) :}

## 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)

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

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

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

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

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