Help about mandelbrot negative exponents

[trenmost]

Hello!

What is the formula used for negative (z --> z^-n + c) rendering?

i applied the following formula:

Code:

RealNumber l = cLength(c1);  //sqrt(c1.r*c1.r+c1.i*c1.i)
 c1.i = sqrt(0.5*(l - c1.r));
 c1.r = sqrt(0.5*(l + c1.r));

but it only gives me a circle...

[element90]

To find the square root of a complex number

if z = a + bi

r = sqrt(a*a + b*b)
theta = atan2(b/a)

z.r = sqrt(r)*cos(theta/2)
z.i = sqrt(r)*sin(theta/2)

[lycium]

Getting the inverse of z^n gives n roots, so element90's code is a little incomplete for the case n=2.

You can solve it quite easily for rational n = p/q (where p and q integers), however dealing with negative p/q is probably also a bit trickier and there might be infinitely many roots...

[claude]

Negative powers are just reciprocals:


Complex reciprocal is defined by:

This is derived by multiplying both the top and bottom of the left hand side by (aka the complex conjugate of ) which leaves the fraction the same.

The escape time algorithm doesn't work for negative powers (there is no possible escape radius, because large values can get small again), see here:
http://math.stackexchange.com/questions/1257555/how-to-compute-a-negative-multibrot-set

Roots are fractional powers, is the n-valued n'th root - the multiple values cause all kinds of problems, not to mention not being analytic at 0 - I wasted some hours trying to find an interior distance estimate for , but the pre-conditions of the Koebe quarter theorem aren't met ( wraps around the unit circle 3 times for that function, for the usual it wraps around just once, which I think is probably crucial).

[DarkBeam]

As far as I can tell it's easy to find the inverse and it's an unique solution.
1/(a+ib) = (a-ib)/((a+ib)(a-ib)) ... etc

[youhn]

See also http://www.fractalforums.com/new-theories-and-research/negative-multibrots/

[xenodreambuie]

The escape time algorithm doesn't work for negative powers (there is no possible escape radius, because large values can get small again), see here:
http://math.stackexchange.com/questions/1257555/how-to-compute-a-negative-multibrot-set

Thanks for that article. Using a more general method for convergent Mandelbrot and Julia formulas, finding fixed critical points by Newton-Raphson fails for z^-n+c. Using c for the critical point solves the problem (although I don't have a smooth transition between these methods).