Fourier Series Graph Interactive

In the graph below, you can add (and remove) terms in the Fourier Series to better understand how it all works.

The examples given on this page come from this Fourier Series chapter.

Our aim was to find a series of trigonometric expressions that add to give certain periodic curves (like square or sawtooth waves), commonly found in electronics and electrical engineering.

On the right of the graph below, as you add terms you'll see the individual sine terms (pink color) appear. (They have been separated vertically so we can see each one clearly.) These individual terms are added to give the blue curve.

Note 1: I'm taking the first "term" to be the constant, `a_0/2`.

Note 2: You can see up to 12 terms.

Things to do

1. Click on the "add term" button to see more terms of the series, what the graph of those terms look like, and the resulting waveform when they are added.
2. Click on the "remove term" button to see less terms
3. Choose either the square, sawtooth or "cos blip" functions and observe the nature of the terms and their graphs.

Function details

Here are the actual functions we examined above.

Square wave

`f(t)={(0, if -4<=t<0),(5, if 0<=t<4):}`

`f(t) = f(t + 8)`

For more information on this example: Full-range Fourier Series - square wave

Sawtooth wave

`f(t)=t, if −pi<=t<pi`

`f(t) = f(t + 2pi)`

For more information on this example: Full-range Fourier Series - sawtooth wave

Cos "blip"

`f(t)={(0,if\ -1<=t<-0.5),(cos 3pit,if\ -0.5<=t<0.5),(0,if\ 0.5<=t<1):}`

`f(t) = f(t + 2)`.