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.

\[ e^{ix} = cos(x) + isin(x) \]

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:

\[ cos(x) = \frac{1}{2} (e^{ix} + e^{-ix}) \]

Likewise, sines can be represented as such too:

\[ sin(x) = \frac{1}{2j} (e^{ix} – e^{-ix}) \]