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##### Fractal Mathematics And New Theories / Re: analytic normals for inflated Mandelbrot set?

« Last post by**gerrit**on

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**Today**at 01:39:40 AMThanks gerrit, it helps a bit, but the expression for \( f(x, y) \) is quite hairy, with part of it being implicit:OK, note that h^2 is the sum of two absolute values of analytic functions, and of course \( \nabla h = \nabla h^2 /(2h) \).

Given complex \( C \) in an atom domain of period \( P \), find \( Z_0 = F_C^P(Z_0) \) where \( F_c(z)=z^2+c \). Now, if \( \left|\frac{\partial}{\partial z}F_C^P(Z_0)\right| \le 1 \) then \( C \) is interior to a component of period \( P \). Now define the surface height

\[ h = f\left(\Re(C), \Im(C)\right) = \frac{ \sqrt{1 - \left| \frac{\partial}{\partial z} \right|^2} }{\left|\frac{\partial}{\partial{c}}\frac{\partial}{\partial{z}} + \frac{\left(\frac{\partial}{\partial{z}}\frac{\partial}{\partial{z}}\right) \left(\frac{\partial}{\partial{c}}\right)} {1-\frac{\partial}{\partial{z}}}\right|} \]

with all the derivatives in that expression being evaluated at \( F_C^P(Z_0) \).

Finding the derivatives of \( f \) seems to be a big challenge, I was wondering if maybe there is a simpler way using properties of conformal mapping etc?

Section 2 of http://persianney.com/fractal/fractalNotes.pdf shows how to go back and forth between \( R^2 \) and \( C \), both for variables and gradient/derivatives.

For analytic function f() consider grad of |f|. In the notes I show that norm of grad is just abs of derivative of f. Similarly you can work out that

\( (\partial_x - i \partial_y)|f| = \partial_z|f| = \bar{f}f^{'}/(2|f|) \). So all you have to work out is \( \partial_c h^2 \) which is only a bit unpleasant.