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''Weierstrass canonical product''
 
''Weierstrass canonical product''
  
An entire function with a given sequence of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201701.png" /> as its zeros. Suppose that the zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201702.png" /> are arranged in monotone increasing order of their moduli, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201703.png" />, and have no limit point in the finite plane (a necessary condition), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201704.png" />. Then the canonical product has the form
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201705.png" /></td> </tr></table>
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$$\prod\left(\frac{z}{\alpha_k},q_k\right)=\prod_{k=1}^\infty W\left(\frac{z}{\alpha_k},q_k\right)=\prod_{k=1}^\infty\left(1-\frac{z}{\alpha_k}\right)e^{P_k(z)},$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201706.png" /></td> </tr></table>
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$$P_k(z)=\frac{z}{\alpha_k}+\frac12\left(\frac{z}{\alpha_k}\right)^2+\ldots+\frac{1}{q_k}\left(\frac{z}{\alpha_k}\right)^{q_k}.$$
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201707.png" /> are called the elementary factors of Weierstrass. The exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201708.png" /> are chosen so that the canonical product is absolutely and uniformly convergent on any compact set; for example, it suffices to take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c0201709.png" />. If the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017010.png" /> has a finite exponent of convergence
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017011.png" /></td> </tr></table>
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$$\beta=\inf\left\lbrace\lambda>0\colon\sum_{k=1}^\infty|\alpha_k|^{-\lambda}<\infty\right\rbrace,$$
  
then all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017012.png" /> can be chosen to be the same, starting, e.g. from the minimal requirement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017013.png" />; this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017014.png" /> is called the genus of the canonical product. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017015.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017016.png" /> diverges for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017017.png" />, then one has a canonical product of infinite genus. The order of a canonical product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020170/c02017018.png" /> (for the definition of the type of a canonical product, see [[#References|[1]]]).
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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.

How to Cite This Entry:
Canonical product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_product&oldid=32945
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article