I am a little bit late with this post but after the previous post I took some time to enjoy it because not every day you can craft a brand new coordinate system…
Also I was a bit in doubt about this post, shall I finally start with those modified Ditichlet kernels or work out a simple de Moivre example in 3D? It became the latter so likely in the next post I will do some first things with modified Dirichlet kernels.
This post is just 4 pictures long (size 550 x 775 pixels) and to be honest it contains no serious math whatsoever. I was only driven by curiosity about how difficult a simple example of the new 3D versions of it would be.
Now it is not 100% trivial but it is also not a very deep result, at best you can say it is a bit technical because you constantly have to apply those sum and difference formulae for the cos and sine functions.
The real deep math work was crafting those exponential circles in 3D in the first place and later finding the coordinate functions belonging to that. That was the deep math because once you have those, the new de Moivre formulae are a piece of cake (make sure it is gluten free!).
In this update I also would like to make an advertisement for a long update I started about one year ago on the other website; all in all it is 37 pictures long (size 550 x 1100) and it took about 3 weeks to write it. It has the title The Missing Equations because with those modified Dirichlet kernel I knew I had solved a terrible hard problem but the higher in the dimensions I got the more missing equations I had for my wonderful solution… Here is the link:
From 14 July 2015: The missing equations.
After this small advertisement here are the four pictures of this update:
Once more: This is not a deep mathematical result or so.
I was just curious of how difficult it would be to get one of the most simple 3D new de Moivre formula results using only the 3 century old stuff from about 50 years before Euler entered the scene…
The most important lesson you can learn from this is that instead of focusing on all kinds of details like n = 2 and only the real part and so on is a waste of time and energy.
After all using the exponential circle is what brings peace to the heart; it is simple, it covers all powers at the same time and you have all coordinate functions at once…
Now you have to wait a few minutes more because I would like to pop up a fresh home brew and after that I will hit the ‘publish post’ button…
Thanks for the waiting 😉 Now I will hit the publish button and see you next time around!