# Understanding entire functions

I am trying to understand the Weierstrass factorization theorum and I was told that the following is true for entire functions...

1. Any finite sequence $$\{c_{n}\}$$ in the complex plane has an associated polynomial p(z) that has zeroes precisely at the points of that sequence, $$p(z)=\prod_{n}(z-c_{n})$$.

2. Also, any polynomial function p(z) in the complex plane has a factorization $$p(z)=a\prod _{n}(z-c_{n})$$, where a is a non-zero constant and $$c_{n}$$ are the zeroes of p.

Can anyone explain this to me in Layman's terms in way that I can understand why these two rules are true, or perhaps just provide a toy example so that I can visualize this better. I understand what complex functions are but I can't seem to understand why these rules hold true.

## 1 Answer

To explain the first one, just take any finite sequence, call the terms $$c_1,c_2,\ldots,c_n$$. The only way a polynomial can equal zero at some point $$z_0$$ is if $$(z-z_0)$$ is a factor of the polynomial. Well, if we know all of the factors of the polynomial, we can just multiply the factors together. Putting this information together, we want a polynomial that vanishes at $$c_1,\ldots, c_n$$ and nowhere else; well, $$z-c_1$$ must be a factor, as must $$z-c_2$$, and so on and so forth. Thus, up to a multiplicative constant, our polynomial must equal $$P(z)=a(z-c_1)(z-c_2)\cdots(z-c_n)=a\prod_{k=1}^n(z-c_k).$$ We need the constant in front because, as long as $$a\neq0$$, the polynomial is still going to have the same zeroes and only these zeroes. Once you impose a condition, such as $$P(1)=6$$, then you can determine an explicit $$a$$.

Your second question is precisely the fundamental theorem of algebra; I think it is most easily expressed as the following:

A polynomial of degree $$n$$ has exactly $$n$$ complex zeros.