[복소해석학] 5.4. Weierstrass infinite product

Theorem 1. (Weierstrass) Given any sequence $\{a_n\}$ of complex numbers with $|a_n|\to\infty$ as $n\to\infty$, there exists an entire function $f$ that vanishes at all $z=a_n$ and nowhere else. Any other such entire function is of the form $f(z)e^{g(z)}$ where $g$ is entire.

Here we need to consider the multiplicities of the zeros. Suppose $\{z_1, z_2,\cdots\}$ is the set of zeros in question where $m_k$ is the multiplicities of $z_k$. Note that the sequence $\{a_n\}$ of prescribed zeros is

$$\{a_n\}=\{z_1, z_1, \cdots(m_1\;\text{times}), z_2, z_2,\cdots(m_2\;\text{times}),\cdots \}$$

We begin the proof with the two entire functions $f_1,\;f_2$ that vanishes at all $z=a_n$ and nowhere else. Then by the above remark, $f_1$ and $f_2$ have same multiplicities at each of their zeros so $f_1/f_2$ has removable singularities at each of its zeros. Hence $f_1/f_2$ is entire and vanishes nowhere. Since $\mathbb{C}$ is a simply connected region, from Section 6 of Chapter 3, [1] we have entire function $g$ such that $f_1(z)/f_2(z)=e^{g(z)}$. Therefore $f_1(z)=f_2(z)e^{g(z)}$ hence the last statement of the Theorem 1. is verified.

To construct the entire function that vanishes exactly at $z=a_n$, we naively suggest the product

$$f(z)=\prod_{n=1}^\infty\left(1-\frac{z}{a_n}\right)$$

However the problem is that this product may not converge. Only with the suitable sequence ${a_n\}$, the product converges. To deal with this problem, we adopt the concept of canonical factors.

Definition. For each integer $k\ge 0$, we define canonical factors by

 

$$E_0(z)=1-z\;\;\;\text{and}\;\;\;E_n(z)=(1-z)\exp\left(z+\frac{z^2}{2}+\cdots+\frac{z^n}{n}\right)$$

Lemma 1. If $|z|\le 1/2$, then $|1-E_k(z)|\le c|z|^{k+1}$ for some $c>0$

proof. If $|z|\le 1/2$ we may write $1-z=\exp(\operatorname{Log}(1-z))$ and then
    \begin{align*}
        E_k(z)&=(1-z)\exp\left(z+\frac{z^2}{2}+\cdots+\frac{z^k}{k}\right)=\exp\left(\operatorname{Log}(1-z)+z+\frac{z^2}{2}+\cdots+\frac{z^k}{k}\right)\\
        &=\exp\left(-\sum_{n=1}^\infty\frac{z^n}{n}+z+\frac{z^2}{2}+\cdots+\frac{z^k}{k}\right)=\exp\left(-\sum_{n=k+1}^\infty\frac{z^n}{n}\right)
    \end{align*}
Now let $w=-\sum_{n=k+1}^\infty z^n/n$ then from $|z|\le 1/2$ we have
\begin{align*}
    |w|\le\sum_{n=k+1}^\infty\frac{|z|^n}{n}\le\sum_{n=k+1}^\infty|z|^n=|z|^{k+1}\sum_{i=0}^\infty|z|^i\le|z|^{k+1}\sum_{i=0}^\infty2^{-i}\le 2|z|^{k+1}
\end{align*}
Then $|w|\le 1$ so we also have

$$|1-E_k(z)|=|1-e^w|\le(e-1)|w|\le c|z|^{k+1}$$

since $|1-e^w|=|\sum_{n=1}^\infty w^n/n!|\le|w|\sum_{n=1}^\infty|w|^{n-1}/n!\le|w|\cdot\sum_{n=1}^\infty 1/n!=(e-1)|w|$.

Suppose we are given a zero of order $m$ at the origin and that $a_1, a_2,\cdots$ are all non-zero. Now we suggest the Weierstrass product by

$$f(z)=z^m\prod_{n=1}^\infty E_n(z/a_n)$$

We claim that $f$ is entire with a zero of order $m$ at the origin, has each $a_n$ as its zeros and vanishes nowhere else. To prove this, we fix $R>0$ and observe the behavior of $f$ in the disc $|z|<R$. If $f$ satisfies those properties and holomorphic in the given disc, since our choice of $R$ was arbitrary, our proof is over.

If $|a_n|\le 2R$, there are only finitely many such terms since $|a_n|\to\infty$. Then the finite product

 

$$z^m\prod_{|a_n|\le 2R}E_k(z/a_n)$$

is holomorphic in the disc $|z|<R$ and vanishes at all $z=a_n$ with $|a_n|<R$. If $|a_n|>2R$ we have $|z/a_n|\le 1/2$ so by the previous lemma,

$$|1-E_n(z/a_n)|\le c|z/a_n|^{n+1}\le c/2^{n+1}$$

Then by Theorem 3.2, Chapter 5, [1],

$$\sum_{|a_n|>2R}|1-E_n(z/a_n)|<\infty$$

hence the product

$$\prod_{|a_n|>2R}E_n(z/a_n)$$

uniformly converges so is holomorphic and does not vanish in the disc $|z|<R$. Thus the function $f$ has the desired properties, which ends the proof of the Weierstrass theorem.

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[1], Stein, Elias M., Shakarachi, Rami 2003. Complex Analysis. Princeton University Press.
[2], 김영원, 계승혁. 2014. 기초복소해석. 서울대학교출판문화원.
[3], 김영원, 2023. 복소해석학 강의노트. 한빛아카데미.
[4], Rudin, Walter 1976. Principles of Mathematical Analysis 3ed. McGraw-Hill Press.
[5], Munkres, J.R., 2000. Topology 2ed. Pearson Education.