# Questions tagged [weierstrass-factorization]

This tag is for questions relating to Weierstrass factorization theorem, an extension of the fundamental theorem of algebra.

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### How to understand exponential in infinite Weierstrass product of $\sin(\pi z)$

Based on the discussion in Understanding infinite product of $sin(\pi z)$, I have a very rudimentary question: How do we derive the equation $\sin(\pi z) = \pi z \prod_{n\neq 0} (1-z/n)e^{z/n}$ in the ...
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### Weierstrass factorization Theorem with simple zeros at Gaussian Integers

I'm studying Weierstrass factorization but I'm stuck on this exercise from Gamelin that asks this: "Construct an entire function that has simple zeros at the Gaussian integers m+ni, − ∞ < 𝑚 , ...
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### Constructing an entire function $f(z)$ such that $f(0)=1$ and $f(n^{1/3}-n^{1/6})=0$ for every $n\geq 2$

I would like to find an entire function satisfying $f(0)=1$ and $f(n^{1/3}-n^{1/6})=0$ for every $n\geq 2$. My initial thought was to use the Weierstrass Factorization Theorem which states if {$a_n$} ...
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### Factorization of modular forms using zeros and poles

In complex analysis, it is possible to decompose any given entire function into a product of linear factors using the zeros, using the Weierstrass factorization theorem. For example, $\sin(z)$ is ...
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### Finding the function by an infinite sum

I have found an infinite sum $$\prod_{n=1}^{\infty}\Big(1-\frac{(n+1)^2x^2}{n^4}\Big)$$ This function acts similarly to the sine function, as shown in the diagram below: This function diverges when ...
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### Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros.

Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros. I know I have to use Weierstrass factorization theorem somehow but I’...
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### Infinite product of trigonometric function [closed]

I would like to find the infinite product of $\frac {\sin x\pi}{\sin \sqrt{x}\pi}$. I have tried to seperate them into two parts namely $$\prod_{n=1}^{\infty}\frac{n^2-x^2}{n^2-x}$$ but it seems to ...
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### Converting equation into short Weierstrass form

I have the curve $y^2 + 4y = x^3 + 3x^2 −x + 1$. I need to find a transformation of the form $X=x+a$ and $Y=y+b$ that turns this curve into the standard form of the elliptic curve: $$Y^2=X^3+AX+B.$$ I ...
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### Can we do Weierstrass factorization for trigonometric substitutions? If so, in which cases?

We have in complex analysis the famous Weierstrass factorization theorem which says that every entire function can be written $$f(z) = z^ne^{g(z)}\prod_{k=0}^\infty E_{p_n} \left(\frac{z}{a_n}\right)$$...
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### Zeroes of a non-zero entire function

Let $f$ be a nonzero entire function such that $|f(z)|\leq e^{|z|}$. Can $f(\sqrt{n})=0$ for infinitely many $n$? We know that the Weierstrass Factor Theorem asserts that given any closed discrete set ...
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