Difference between revisions of "Product of Weierstrass elementary factors is entire"
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(Created page with "==Theorem== Let $\{a_n\}$ denote a sequence of complex numbers with $a_n \neq 0$ and $\displaystyle\lim_{n\rightarrow\infty} |a_n| = \infty$. If $\{p_n\}$ i...") |
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$$\displaystyle\sum_{k=1}^{\infty} {r \choose |a_k|}^{1+p_n} < \infty,$$ | $$\displaystyle\sum_{k=1}^{\infty} {r \choose |a_k|}^{1+p_n} < \infty,$$ | ||
where ${r \choose r_k}$ denotes a [[Binomial coefficient]], then the product | where ${r \choose r_k}$ denotes a [[Binomial coefficient]], then the product | ||
| − | $$P(z) = \displaystyle\prod_{k=1}^{\infty} E_{p_k} \left( \dfrac{z}{a_k} \right)$$ | + | $$P(z) = \displaystyle\prod_{k=1}^{\infty} E_{p_k} \left( \dfrac{z}{a_k} \right),$$ |
| − | defines an [[entire]] function which has a zero at each $a_k$ and no other zeros. If a complex number $w$ appears $n$ times in the sequence $\{a_n\}$, then $P$ has a zero of [[order of a zero|order]] $m$ at $w$. | + | where $E_{p_k}$ denotes a [[Weierstrass elementary factors|Weierstrass elementary factor]] defines an [[entire]] function which has a zero at each $a_k$ and no other zeros. If a complex number $w$ appears $n$ times in the sequence $\{a_n\}$, then $P$ has a zero of [[order of a zero|order]] $m$ at $w$. |
==Proof== | ==Proof== | ||
Latest revision as of 19:29, 26 November 2016
Theorem
Let $\{a_n\}$ denote a sequence of complex numbers with $a_n \neq 0$ and $\displaystyle\lim_{n\rightarrow\infty} |a_n| = \infty$. If $\{p_n\}$ is a sequence of nonnegative integers such that for every $r>0$ $$\displaystyle\sum_{k=1}^{\infty} {r \choose |a_k|}^{1+p_n} < \infty,$$ where ${r \choose r_k}$ denotes a Binomial coefficient, then the product $$P(z) = \displaystyle\prod_{k=1}^{\infty} E_{p_k} \left( \dfrac{z}{a_k} \right),$$ where $E_{p_k}$ denotes a Weierstrass elementary factor defines an entire function which has a zero at each $a_k$ and no other zeros. If a complex number $w$ appears $n$ times in the sequence $\{a_n\}$, then $P$ has a zero of order $m$ at $w$.
