Welcome to Fractal Forums

Hey guys,

I'm usually not very active in these forums, but recently I have had an idea that I must share with you.


First I took this amazing equation:
(http://latex.codecogs.com/png.download?e^{ix}&space;=&space;\cos&space;x&space;+&space;i&space;\sin&space;x&space;\newline)
and knowing this property:
(http://latex.codecogs.com/png.download?i^4&space;=&space;1&space;\;\;;\;\;&space;i^3&space;=&space;-i&space;\;\;;\;\;&space;i^2&space;=&space;-1&space;\;\;;\;\;&space;i^1&space;=&space;i&space;\;\;;\;\;&space;i^0&space;=&space;1)
and the successive derivatives of the cos/sin:
(http://latex.codecogs.com/png.download?\frac{\mathrm{d}^4&space;}{\mathrm{d}&space;x^4}&space;(\sin&space;x)&space;=&space;\frac{\mathrm{d}^3&space;}{\mathrm{d}&space;x^3}&space;(\cos&space;x)&space;=&space;\frac{\mathrm{d}^2&space;}{\mathrm{d}&space;x^2}&space;(-\sin&space;x)&space;=&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}&space;(-\cos&space;x)&space;=&space;\sin&space;x)


I noticed that the powers of (http://latex.codecogs.com/png.download?i) and the derivatives of cos/sin both repeat after 4 iterations.


Next I defined (http://latex.codecogs.com/png.download?h) as having this property:
(http://latex.codecogs.com/png.download?e^{hx}&space;=&space;\cosh&space;x&space;+&space;h\sinh&space;x)
Because I knew that :
(http://latex.codecogs.com/png.download?\frac{\mathrm{d}^2&space;}{\mathrm{d}&space;x^2}&space;\sinh&space;x&space;=&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}&space;\cosh&space;x&space;=&space;\sinh&space;x)
And after satisfying rules of exponentiation, it turns out, amazingly:
(http://latex.codecogs.com/png.download?h^2&space;=&space;1&space;\;\;;\;\;&space;h^1&space;=&space;h&space;\;\;;\;\;&space;h^0&space;=&space;1)


Again I notice that the powers of (http://latex.codecogs.com/png.download?h) and the derivatives of cosh/sinh both repeat after 2 iterations.


Now I want define a (http://latex.codecogs.com/png.download?j) and find a (http://latex.codecogs.com/png.download?f(x)) such that:
(http://latex.codecogs.com/png.download?e^{jx}&space;=&space;f'(x)&space;+&space;jf(x))
where (http://latex.codecogs.com/png.download?f(x)) satisfies:
(http://latex.codecogs.com/png.download?\frac{\mathrm{d}^6&space;}{\mathrm{d}&space;x^6}&space;f(x)&space;=&space;f(x))
so that we may get:
(http://latex.codecogs.com/png.download?j^6&space;=&space;1&space;\;\;\textrm{and}\;\;&space;j^3&space;=&space;-1)


Thus the powers of (http://latex.codecogs.com/png.download?j) and the derivatives of (http://latex.codecogs.com/png.download?f(x)) would both repeat after 6 iterations. I believe that (http://latex.codecogs.com/png.download?j) would then be a true extension to imaginary numbers. Using WolframAlpha: http://www.wolframalpha.com/input/?i=f%27%27%27%27%27%27+%3D+f (http://www.wolframalpha.com/input/?i=f%27%27%27%27%27%27+%3D+f) we find some potential (http://latex.codecogs.com/png.download?f(x))

This process can be continued to get (http://latex.codecogs.com/png.download?k,&space;\;l,&space;\;m,&space;\;n,\;\ldots) and completely generalize complex numbers to higher dimensions.


What are your thoughts on this? For me this is very elegant way to extend complex numbers. My next steps are to find the function that satisfies those properties and to see if this has some more interesting properties. Maybe even test for a true 3D mandelbrot  ;D

I really appreciate your help.

Well, that's an interesting property... but certainly no extension of the complex numbers. You can solve your problem by observing that h is actually equal to -1, a square root of 1. Similarly, i and -i are fourth roots of 1. So to solve the case of 6 iterations, you need j to be a sixth root of 1, for instance exp(pi i/3). Then take
f(x) = 1/2(exp(jx) + exp(-jx))
It will solve your differential equation (although it's not the only solution, of course).

Best,

Sam

P.S.: How do you type latex equations on this board?