# Expansion of Complex Functions

I apologize for my unusual terminology, but my math training in this field is rather lacking, and not entirely in English.

Consider the set of complex functions that are holomorphic except for a finite or countably infinite set of isolated singularities (either poles or essential singularities). I break these functions up into tiers, where the tier can be written in shorthand as a subscript of T followed by the level tier. Let a "tier-0" function be a nonzero constant function. Let a "tier-1" function be any complex function that can be written as $$f_{T1}(z)=f_{T0}\prod_i(1-\frac{z}{\sigma_i})^{\mu_i}$$, where $$\mu_i$$ is the multiplicity of a zero $$\sigma_i$$ if it is positive, and is the order of a pole $$\sigma_i$$ if it is negative. Let a "tier-2" function be any complex function that can be written in the form $$f_{T2}(z)=f_{T1}e^{\prod_j(1-\frac{z}{\sigma_j})^{\mu_j}}$$. Let a "tier-3" function be any complex function that can be written in the form $$f_{T3}(z)=f_{T2}e^{e^{\prod_k(1-\frac{z}{\sigma_k})^{\mu_k}}}$$, and so on for higher tiers.

Are the following statements correct:

1) The sum or product of a tier A and a tier B function, where $$A>B$$ is always going to be a tier A function. The sum or product of two tier A functions is a tier B function, where $$A\geq B$$.

2) The derivative of a tier A function for $$A\geq 2$$ will always be a tier A function. The derivative of a tier 1 function can be either a tier 1 or a tier 0 function.

3) The integral of a tier A function for $$A\geq 1$$ will always be a tier A function.

4) Every complex function that is analytic everywhere except for a finite or countably infinite number of isolated singularities can be expressed as a tier A function for some finite tier A.

EDIT: For each of the product notations, if $$\sigma_n=0$$, then the term in the product is $$\sigma_n^{\mu_n}$$

• Too many questions perhaps, and also is there any reason you are asking this? Sep 15, 2019 at 18:40
• @SimplyBeautifulArt Regarding the multiple questions, statement 3 is really equivalent to statement 2; if statement 1 is false then statements 2 and 3 must necessarily also be false. One could, in theory, answer whether or not all three statements are correct with one proof or counterexample. Regarding why I am asking, I was looking at this tier system as a possible alternative to the generalization of Weierstrass's factorization theorem. Sep 15, 2019 at 18:50