using power and sum rule in differentiation one gets:

Z' = dFc / dZ = F'

because Z and Z' are complex numbers so :

Re( Z' ) = Z'.X = 2 * Z.X

Im (Z' ) = Z'. Y = 2 * Z.Y

in Maxima:

(%i3)P(n):=if n=0 then z else P(n-1)^2+c;

(%o3) P(n):=if n=0 then z else P(n-1)^2+c

(%i4) diff(P(1), z);

(%o4) 2*z

Applications:

It can be stated in any of several equivalent forms:

dF

dF

F'

definition of function in pseudocode:

F

begin

Z

For i=1 to n do

Z

return Z

end;

-------------------------------

Z'

---------------------------------

It goes like that :

Z

Z'

Z

Z'

Z

Z'

Z

Z'

...

Z

Z'

and so on ...

-----------------------------

How it was computed ?

First some notes:

1.folding a function:

F

example:

F

Here one may find usage of words : itereation, recursion

in the same meaning while in the programming theory thay have diffrent meaning.

2.it is a recursive definition of function

Z

3.Lets write definition of map

Z

in a diffrent form :

Z

Here G(c) is a composite function

G(c):=E(I(c))

which consisist of external function E(u) and internal function I(c):

E(u)=u

u:=F

so:

E':=dE/du:=2*u:=2*Z

I(c):= F

I'(c):=dI/dc := Z'

(Here one do not compute Z'

To compute derivative of composite function G(c) one must apply chain rule

dG/dc:=dE/du * dI/dc

so:

G'(c)=dG/dc = E'(u) * I'(c) = (2*Z

Tu compute derivative of map:

Z

which is a sum of 2 functions use sum rule in differentiation:

F'(c)= F

here H(c)=c

It is a monic polynomial of degree=1 ( (:-))) = linear function

To compute derivative of H(c) use The power rule:

H'(c)=dH/dc= 1

so

Z'

--------------------------------------------

Z and Z' are complex numbers:

Z = ZX + ZY *i

Z' = ZX' + ZY'*i

so using rules for Addition and Multiplication of complex numbers one gets:

ZY

ZX

ZX'

ZY'

------------------------------------------------

-------------------------------------------------------

Applications of first derivative:

- DEM/M
- locating all the points of bifurcation from a parent to children component in Newton Method
- locate all of the descendants of any given mu-atom
- estimation of distance from component point to the center or the edge of component.
- coloring inside of M-set
- finding multiplier ( = eigenvalue ) of periodic point and internal angle of hyperbolic component of Mandelbtrot set

Z''

See:

- Internal angles and multipliers by Michael Frame, Benoit Mandelbrot, and Nial Neger
- Linas Vepstas
- Mu-ency by Robert P. Munafo
- Wiki
- "Critical Points" for non-analytic functions
- Calculus by Gilbert Strang from MIT
- sci-math discussion
- pl.sci.matematyka
- Dynamics And Hierarchies Daniel Geisler

Autor: Adam Majewski adammaj1-at-o2-dot--pl

Feel free to e-mail me. (:-))

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