Right. Just installed MathML block for WordPress because I want to talk about Fourier Series.
I’m just going to write a quick sidetrack required for the next post (and also for testing the MathML block).
Here’s the Euler’s formula discovered in 1748.
eix=cos(x)+isin(x)
This can be visualized as a unit complex number that rotates clockwise around the origin on the complex plane for every 2π increase in x.
We can reverse the direction of the rotation simply by doing:
e−ix=cos(x)–isin(x)
With this, we can easily represent any cosine as:
cos(x)=12(eix+e−ix)
Likewise, sines can be represented as such too:
sin(x)=12j(eix–e−ix)