Weierstrass Factorization Theorem, infinite polynomial/infinite power series As we know from basic complex analysis, any finite polynomial (infinite power series) $P(z)$ can be represented as a product of its zeroes. $P(z)=\Pi_n(z-z_n)$ (when $z_n $ are the zeros).

And as we know, the "Weierstrass Factorization Theorem" is a super inclusion of the theorem that I mentioned before, and it is pretty hard to prove the Weierstrass Factorization Theorem.
I tried to prove the Weierstrass Factorization Theorem when the function is an infinite power series, but  it seems more complicated than I thought.

Does anyone know if there is a nice (or not) proof to Weierstrass Factorization Theorem when the function is an infinite power series?

It's a bit complicated.
Power series
$$P(z) = \sum_{k=0}^{\infty} c_kz^k$$ and the more general Laurent series $$L(z) = \sum_{k=-\infty}^{\infty} c_kz^k$$ have a radius of convergence. Their values outside of this radius are not defined.
In analysis of one complex variable we learn that we can always construct a power series or Laurent series around a point $z_0$ so that its radius of convergence precisely touches the nearest pole.
A famous example we learn in calculus (one real variable) that the McLaurin series of $\arctan(t)$ converges on $[-1,1]$ on the real line. In one complex variable we can prove that this happens because

*

*center point is $z=0$

*$\arctan(z) = \frac{i}2 \left(\log(1-iz) - \log(1+iz)\right)$

*closest poles to $z=0$ are $z=\pm i$

*$\|\pm i\| = 1$
Now any partial sum of power series will have as many zeroes as it's degree and we can factor any such partial sum with Weierstrass Factorization theorem.
But partial sums to power series are not unique.
Because of the fundamental theorem of Algebra, somewhere outside of the radius of convergence there will be hopping around excess zeroes in some pattern for our choice of partial sum as we increase it's number of terms.
