0
votes
1answer
44 views

Proof of “factorization of polynomials” using only Complex Analysis

I ask for the proof of the following: If $p$ is a polynomial with degree $n\ge 1$ and zeros in $A\subseteq \mathbb C$ whose order (multiplicity) is given by $n:A\to \mathbb N^*$ then $A$ is finite ...
4
votes
1answer
143 views

Is my proof correct? (the product $\prod_{n=1}^\infty (1+\frac{z}{n} ) \mathrm{e}^{-\frac{z}{n}}$ converges absolutely and uniformly on compact sets.)

I want to prove that the product $$\prod_{n=1}^\infty \left(1+\frac{z}{n} \right) \mathrm{e}^{-\frac{z}{n}}$$ converges absolutely, and uniformly on compact subsets of $\mathbb C$: My book ...
10
votes
1answer
308 views

How to prove that $\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}(1-\frac{x^2}{n^2})$ [duplicate]

How to prove that $$\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$$ I tried it with the Taylor series of $\sin(x)$ but I failed. Is there any help?
2
votes
1answer
69 views

proving than an infinite product defines an entire function

Consider the infinite product $$F(z)=\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$ How can i prove that $F$ is entire? Can i write $F$ as a Weierstrass product $\prod ...
2
votes
1answer
245 views

Convergence of infinite product

This could be something which is already somewhere in the website, but I am unable to locate any. Prove $$\prod_{n=1}^{\infty} (1-z^n)$$ converges absolutely and uniformly on each compact subset of ...
5
votes
1answer
127 views

Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$

Difficult question from some test somewhere (I forget). $$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$ $x$ is, of course, an integer.
12
votes
2answers
751 views

Infinite products - reference needed!

I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to ...
7
votes
1answer
135 views

Zeros in the complex plane and convergence

I'm doing some number theory which requires some work in $\mathbb{C}$, but unfortunately my complex analysis is a little rusty. A text I am reading states the following: ...and given that ...
1
vote
1answer
171 views

simplifying a product formula (similar to Euler's sine product)

Can anyone help me out trying to simplify the left hand side of the below equation to obtain the right hand side? $$ \displaystyle\prod_{\substack{n=-\infty \\n\neq ...
5
votes
1answer
112 views

Modulus of infinite product of complex functions

We know that for complex (entire) functions $f,g$ we have $|f(z)g(z)|=|f(z)||g(z)|$, where $|.|$ means complex modulus. What about if we have an infinite product? Is it true that $$\bigg| ...