• May 08, 2021, 05:36:11 PM

### Author Topic:  strange shape for especially complex function sums  (Read 427 times)

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#### hgjf2

• Fractal Feline
• Posts: 169
##### strange shape for especially complex function sums
« on: July 03, 2019, 08:28:17 AM »
I looked that sum series (z+z^4+z^9+z^16+z^25+...z^(n^2)) whick isn't result of integrals or diferential equations has a strange fractal complex map simmilar with Lavaur disk.

Fractal researcher

#### hobold

• Fractal Frogurt
• Posts: 430
##### Re: strange shape for especially complex function sums
« Reply #1 on: July 03, 2019, 11:33:46 AM »
So that function does not converge outside the unit disk, and therefore the graph is black there?

#### claude

• 3f
• Posts: 1850
##### Re: strange shape for especially complex function sums
« Reply #2 on: July 03, 2019, 01:54:19 PM »
it's a polynomial so converges everywhere, but its magnitude is enormous outside the unit disc so probably overflows to Infinity or NotANumber.

edit: but probably the corresponding infinite series doesn't converge outside the unit disc indeed
« Last Edit: July 03, 2019, 02:22:20 PM by claude, Reason: series »

#### hgjf2

• Fractal Feline
• Posts: 169
##### Re: strange shape for especially complex function sums
« Reply #3 on: July 05, 2019, 10:05:27 AM »
Outside disk in NothANumber because is divergent functions like in Riemann Zeta function when takes values a+bi when a<1. And this program don't render divergent to determining the core of logarithmic spiral tangent on set of finit sum.

#### fractower

• Fractal Phenom
• Posts: 54
##### Re: strange shape for especially complex function sums
« Reply #4 on: July 06, 2019, 10:56:20 PM »
I tried sticking in some small Mandelbrot polynomials. It is interesting to compare them.

Iteration 1 = z
Iteration 4 = (((z)^2+z)^2+z)^2+z

The following code produces Mandelbrot polys.

Code: [Select]
#!/usr/bin/perlif(($#ARGV != 0)||($ARGV[0]<1)){    printf("usage: brot_poly.pl N\n");    printf("output: Nth iteration of the Mandelbrot poly (N>=1).\n");      die();}$I =$ARGV[0];for($i=2;$i<=$I;$i++){    printf("(");}printf("z");for($i=2;$i<=$I;$i++){    printf(")^2+z");}printf("\n");

#### hgjf2

• Fractal Feline
• Posts: 169
##### Re: strange shape for especially complex function sums
« Reply #5 on: July 07, 2019, 11:23:20 AM »
This math phenomenon that's related here. That at my math researches if I finding new fractal, I don't hesitating to "seize" it on FRACTALFORUMS.
Also my graphs prooves that the solutions of complex equation sum[k=1 to infinity](z^(k^2))=p are placed into fractal set in complex space, with Hausdorff (fractalish) dimension between 0 and 1, where (p) is a number complex something.

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