closed form for sequence z=z^2+c & generating function
hi there... hope this is in the right place, even guidance to what the particular area of maths im talking about is would be appreciated, so I can find material/books online. thanks. where can I read about this?!
(I've only been learning about power series & generating functions for a week or 2..so this is probably a silly question)
I'm wondering if there's a generating function, and/or closed form (I think it's called) expression for the series z=z^2+c (ie as used in Mandelbrot set calculations).. ie a0=0, a1=c, a2=c^2+c, a3=(c^2+c)^2+c etc.... (z and c are complex)
ie I want to find the value of the nth term directly.
whether using a(n+1)=an^2+an-a(n-1)^2 is any better than a(n+1)=an^2+c, i have no idea... don't know if it's even possible.
also (how similar this question is, not sure..) the same calculation but just using real numbers - ie two sequences, where
and X0=0, Y0=0, a+ib is the c from above.
ie generating functions & closed form expressions for these joined sequences, and pointers to where I could read about this sort of stuff! nothing I've found so far handles such series. thanks very much.
Re: closed form for sequence z=z^2+c & generating function
a closed form for the recursively defined (this is the term you are looking for) mandelbrot sequence is not known. this is often the case with recursively defined functions, so it is a cause for celebration when closed forms are found.
because of this difficulty, it is sometimes quite a chore to determine what the "regions of convergence" are. as the mandelbrot set demonstrates so stunningly, the nature of these sets can be quite complex, even if they admit of a simple description. for these kinds of tasks, a high-speed computer is your friend....
to give another example, the value to which "Newton's method" converges (if it does) depends on an initial guess (a choice of intial value). the boundaries of the regions of convergence, here as well, display a similar kind of behavior as the madelbrot set does, with large regions of "stable" behavior, and unpredictable behavior along the boundaries.
to answer your question about the "real version" of the madelbrot generating function, to get an idea of the behavior there, you might want to look here, and learn more about Julia sets (based on re-iterated polynomials).