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