Difference between revisions of "Canonical product"
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''Weierstrass canonical product'' | ''Weierstrass canonical product'' | ||
− | An entire function with a given sequence of complex numbers | + | 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 | where | ||
− | + | $$P_k(z)=\frac{z}{\alpha_k}+\frac12\left(\frac{z}{\alpha_k}\right)^2+\ldots+\frac{1}{q_k}\left(\frac{z}{\alpha_k}\right)^{q_k}.$$ | |
− | The | + | 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 | + | 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 [[#References|[1]]]). |
====References==== | ====References==== |
Revision as of 19:55, 14 August 2014
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
$$P_k(z)=\frac{z}{\alpha_k}+\frac12\left(\frac{z}{\alpha_k}\right)^2+\ldots+\frac{1}{q_k}\left(\frac{z}{\alpha_k}\right)^{q_k}.$$
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.
Canonical product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_product&oldid=32945