10/10/2014, 11:51 PM
If an initial value x converges to a fixpoint at an angle y , we can give that value pair(x,y) a corresponding color depending on the angle y.
We can make a plot that would then be the analogue of a fractal , the coloring based on the values y corresponding to the x's.
---
That is the main idea ...
Some comments : the derivative at the fixpoints x_0 needs to be positive real.
SO :
Arg ( f ' (x_0) ) = 0
0 < Abs ( f ' (x_0) ) < 1
Also the case is simplest when f(z) only has 2 conjugate fixpoints.
***
Therefore I seek :
Taking - without loss of generality ? - the fixpoints equal to +/- i .
g(z) = real entire = ??
f(z) :=
exp(g(z)) (z^2+1) + z
Arg ( f ' (i) ) = 0
0 < Abs ( f ' (i) ) < 1
And also f(z) ~ exp(z) z^A for some small real A.
Many solutions g(z) must exist , but which is best ?
And how do the " new fractals " look like ?
---
Also can the angle be given by an integral ?
Contour integral ?
z_0 = initial value
z_n = f ( z_(n-1) )
lim n -> +oo
[z_n - z_(n-1)] / [ | z_n - z_(n-1) | ]
= contour integral ( z_0 ) ???
regards
tommy1729
We can make a plot that would then be the analogue of a fractal , the coloring based on the values y corresponding to the x's.
---
That is the main idea ...
Some comments : the derivative at the fixpoints x_0 needs to be positive real.
SO :
Arg ( f ' (x_0) ) = 0
0 < Abs ( f ' (x_0) ) < 1
Also the case is simplest when f(z) only has 2 conjugate fixpoints.
***
Therefore I seek :
Taking - without loss of generality ? - the fixpoints equal to +/- i .
g(z) = real entire = ??
f(z) :=
exp(g(z)) (z^2+1) + z
Arg ( f ' (i) ) = 0
0 < Abs ( f ' (i) ) < 1
And also f(z) ~ exp(z) z^A for some small real A.
Many solutions g(z) must exist , but which is best ?
And how do the " new fractals " look like ?
---
Also can the angle be given by an integral ?
Contour integral ?
z_0 = initial value
z_n = f ( z_(n-1) )
lim n -> +oo
[z_n - z_(n-1)] / [ | z_n - z_(n-1) | ]
= contour integral ( z_0 ) ???
regards
tommy1729