About periodical complex functions

[hgjf2]

Let a function f:C->C as periodical tiling function (exist k,l real number with f(z)=f(z+ik+jl), whatever i,j integer numbers) as exemplu
 f(z) = integral(sqrt(1/sin(z)))dz. I looked that exist a least one z0 at whick f(z0) = oo.
 

[hgjf2]

Let a function f:C->C as periodical tiling function (exist k,l real number with f(z)=f(z+ik+jl), whatever i,j integer numbers) as exemplu
 f(z) = integral(sqrt(1/sin(z)))dz. I looked that exist a least one z0 at whick f(z0) = oo.
 

Demonstration
Whatever f:C->D<C a complex function, normaly that D is strictly whole C complex numbers set.
If define this periodical complex function f:C->C as periodical function f(z) = f(z+ik+jl) whatever i,j integer and k,l two real number defined same as pi at sinus function and exponential function then (f) can be define as f:D<C ->C where D is a rectangle what have dimensions (k) and (l) then the invert function g=f[-1] normaly f(D) must include whole C because not exist complex functions holomorphs f:D1->D2 at whick the domains D1,D2<C isn't whole C

[hgjf2]

If make fractals Julia from the periodic complex functions f(z)=f(z+ik+jl) , those fractals will filling space with Julia sets exacthly same at the pictures from topic "Space filling Julia set" here in this chapter "new theory and researchs"

[s31415]

I comfess I have trouble understanding what you mean exactly. But one way of making periodic Julia pattern is to use the Weierstrass elliptic function, see this blog post:
http://algorithmic-worlds.net/blog/blog.php?Post=20120330
The main point is that this function is periodic AND conformal, so it does not distort Julia patterns. It does have poles but this is not a problem.

Sam

[Pauldelbrot]

It's true that a nonconstant meromorphic elliptic function will have poles in every tile of repetition. (In fact, the sum of multiplicities of poles and the sum of multiplicity of zeroes will be equal.)

However, I'm pretty sure you can get attracting basins with such functions. Poles will have a mosaic of copies of the whole fractal "draining down them" and won't therefore be in an attracting basin (so instead the Julia set) unless every point goes to a single attractor. The conditions do exist (infinity not an attractor, poles exist) for some parametrization of an elliptic function to have parameter values for which there are no attractors, though.