Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the product $$\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$

I know that this product defines an entire function $F$. I must show that the order of growth of $F$ is finite, at most 2. My definition of order of an entire function is $$\rho=\inf\{\lambda>0: \displaystyle\sup_{|z|=r} |f(z)|=O(e^{r^{\lambda}}),r\rightarrow\infty\}$$ Can someone give me an help?

share|improve this question
1  
Maybe Jensen's formula could be of help: en.wikipedia.org/wiki/Jensen%27s_formula –  Eric Haengel Feb 18 '13 at 21:27

1 Answer 1

The zeros of $F$ are $n+ki$ for $n=1,2,\dots$, $k\in\mathbb{Z}$. If $p>2$ then $$ \sum_{n=1}^\infty\sum_{k=-\infty}^{\infty}\frac{1}{|n+ki|^p}<\infty. $$ This shows that the order of $F$ is at most $2$.

share|improve this answer
    
sorry, i can't understand. What i knew was that the convergence of the double series you wrote tells me that the convergence exponent of zeros of $F$ is 2, but the order of $F$ is at least the convergence exponent of zeros, not at most....what is wrong in what i know? –  Federica Maggioni Feb 19 '13 at 7:51
    
Nothing. I should not answer questions after midnight. –  Julián Aguirre Feb 19 '13 at 18:13

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.