Fourier Series: Several functions

Second post dedicated to the Fourier series [Link] with several examples of classical periodic functions.


1- Triangle wave

The formula is:

$f\left(x\right)=\frac{4}{\pi }(\frac{1}{9}\mathrm{cos(3x)}+\frac{1}{25}\mathrm{cos(5x)}+...+\frac{1}{n^2}\mathrm{cos(}nx))$

Or can be written like this...

$f\left(x\right)=\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{\mathrm{cos(2k+1)}.x}{{\mathrm{(2k+1)}}^2}$

and the corresponding macro...

+++ IJ snippet: Triangle_wave_fourier_series.ijm +++
+++ End of IJ snippet +++

 To create the triangle wave, select the whole image (Ctrl+A) then compute the profile (Analyze > Plot Profile or Ctrl +K).

Fig. 1: Triangle wave obtained from the sum of each row of the Fourier Series.

2- Sawtooth wave

Another function defined by ...

The following video shows the evolution of the resulting curve when we add more components.

3- JavaScript

Here is a small script containing all the various functions previously described. For sake of convenience, this script is written in JavaScript but uses exactly the same function Process > Math > Macro...

+++ IJ JavaScript snippet +++ +++ End of IJ JavaScript snippet +++

4- Links

Square wave image [Link]