Helpful Revision for Fourier Series
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.
On this page :
Properties of Sine and Cosine Graphs
Periodic Functions
Continuity
Split Functions
Summation Notation
Useful Integrals
Properties of Sine and Cosine Functions
These properties can simplify the integrations that we will perform later in this chapter.
The Cosine Function
Background
From previous chapters:
Sine and cosine curves
Even and odd functions
Integral of Sine and Cosine
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)
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)
Multiples of π for Sine and Cosine Curves
Consider the function y = sin x
From the graph (or using our calculator), we can observe that:
| sin(nπ) = 0 | for n = 0, 1, 2, 3, ... (in fact, all integers) |
 |
for n = 0, 1, 2, 3, ... (in fact, all integers) |
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:
(a) f(t) = sin t.
Useful Background
For f(t) = sin t, we have: f(t) = f(t + 2π). The period is 2π.
(b) Saw tooth waveform, period = 2:
Useful background
For this function, we have:
f(t) = 3t (for -1 ≤ t < 1)
f(t) = f(t + 2) [This indicates it is periodic with period 2.]
(c) Parabolic, period = 2.
Useful background
For this function, we have:
f(t) = t2 (for 0 ≤ t < 2)
f(t) = f(t + 2) [Indicating it is periodic with period 2.]
(d) 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
(a) 
(b) 
Useful Background
(c) 
(d) 
Summation Notation
It is important to understand summation notation when dealing with Fourier series.
Examples
Expand the following and simplify where possible:
1. 
2. 
3. 
4. 
Some Useful Integrals
These are obtained from integration by parts:
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