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Welcome to Fractal Forums > Fractal Software > Programming > Help about mandelbrot negative exponents
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Author Topic: Help about mandelbrot negative exponents  (Read 355 times)
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trenmost
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Posts: 3
« on: April 30, 2015, 08:28:00 AM »
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...
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element90
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« Reply #1 on: April 30, 2015, 02:07:24 PM »
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)
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lycium
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« Reply #2 on: April 30, 2015, 02:16:58 PM »
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...
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claude
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« Reply #3 on: April 30, 2015, 03:16:09 PM »
Negative powers are just reciprocals:
z^{-n} = \frac{1}{z^n}

Complex reciprocal is defined by:
\frac{1}{x + y i} = \frac{x - y i}{x^2 + y^2}
This is derived by multiplying both the top and bottom of the left hand side by x - y i (aka the complex conjugate of x + y i) 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, z^{\frac{1}{n}} 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 z \to z^{-2} + c, but the pre-conditions of the Koebe quarter theorem aren't met (\frac{\partial}{\partial z} f^p(z_0(c), c) wraps around the unit circle 3 times for that function, for the usual z^{+2} + c it wraps around just once, which I think is probably crucial).
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DarkBeam
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« Reply #4 on: April 30, 2015, 04:29:22 PM »
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 wink
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youhn
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« Reply #5 on: April 30, 2015, 05:22:21 PM »
See also http://www.fractalforums.com/new-theories-and-research/negative-multibrots/
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xenodreambuie
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« Reply #6 on: May 03, 2015, 12:28:28 AM »
Quote from: claude on April 30, 2015, 03:16:09 PM

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).
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