Canonical product

Weierstrass canonical product

An entire function with a given sequence of complex numbers $\{\alpha_k\}$ as its zeros. Suppose that the zeros $\alpha_k\neq0$ are arranged in monotone increasing order of their moduli, $|\alpha_k|\leq|\alpha_{k+1}|$, and have no limit point in the finite plane (a necessary condition), i.e. $\lim_{k\to\infty}\alpha_k=\infty$. Then the canonical product has the form

$$\prod\left(\frac{z}{\alpha_k},q_k\right)=\prod_{k=1}^\infty W\left(\frac{z}{\alpha_k},q_k\right)=\prod_{k=1}^\infty\left(1-\frac{z}{\alpha_k}\right)e^{P_k(z)},$$

where

$$P_k(z)=\frac{z}{\alpha_k}+\frac12\left(\frac{z}{\alpha_k}\right)^2+\dotsb+\frac{1}{q_k}\left(\frac{z}{\alpha_k}\right)^{q_k}.$$

The $W(z/\alpha_k,q_k)$ are called the elementary factors of Weierstrass. The exponents $q_k$ are chosen so that the canonical product is absolutely and uniformly convergent on any compact set; for example, it suffices to take $q_k\geq k-1$. If the sequence $\{|\alpha_k|\}$ has a finite exponent of convergence

$$\beta=\inf\left\lbrace\lambda>0\colon\sum_{k=1}^\infty|\alpha_k|^{-\lambda}<\infty\right\rbrace,$$

then all the $q_k$ can be chosen to be the same, starting, e.g. from the minimal requirement that $q_k=q\leq\beta\leq q+1$; this $q$ is called the genus of the canonical product. If $\beta=\infty$, i.e. if $\sum_{k=1}^\infty|\alpha_k|^{-\lambda}$ diverges for any $\lambda>0$, then one has a canonical product of infinite genus. The order of a canonical product $\rho=\beta$ (for the definition of the type of a canonical product, see [1]).

References

[1]	B.Ya. Levin, "The distribution of zeros of entire functions" , Amer. Math. Soc. (1980) (Translated from Russian)

Comments

See also Blaschke product; Entire function; Hadamard theorem.