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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Reducing polynomial from Weierstrass substitution from n-degree to n/2-degree
If you have an 8th degree polynomial $f(t)$, and you know that despite 8 solutions, there are 4 solutions $x_0,x_1,x_2,x_3$ but the other 4 solutions are $2*\arctan(x0)+\pi, 2*\arctan(x1)+\pi, 2*\...
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How do I "stabilize" the interpolator that uses the roots?
In a previous question, I have been instructed that, in order to pass through all and only the distinct points $(x_{1}, 0) ... (x_{k}, 0)$ with a non-constant continuous function, I can simply use ...
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Polynomial or wave through aligned points [closed]
If I have the distinct points $(x_{1}, 0) ... (x_{n}, 0)$,
A) What would be a simple polynomial (non-constant) passing through these points? And could I instead also write a simple continuous curve (a ...
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Use Weierstrass Factorization Theorem to prove $\sinh \pi z=\pi z \prod_{n=1}^{\infty}(1+\frac{z^2}{n^2})$
Here is my idea.
$\sinh \pi z=\frac{1}{2}(e^{\pi z}-e^{-\pi z})$, its zeros are $a_n=ni$, and $a_0$ is a zero of order $1$. Since $\forall R>0,$ $\sum_{n=1}^{\infty}\left(\frac{R}{|a_n|}\right)^2$ ...
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Ash - Complex Variables - Sufficient conditions for the absolute and uniform convergence of the infinite product of functions
I am self studying Ash & Novinger's Complex Variables. The authors prove the following theorem (see Page 4, Subsection 6.1.6):
Proposition. Let $g_1 , g_2, \ldots$ be a sequence of bounded ...
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An entire function that only has simple zeroes at the positive squares
By the Weierstrass Factorization Theorem, every entire function f can be represented as a product involving its zeroes.
Moreover, if $\{ a_n \}$ is the sequence of zeroes of $f$ then $\displaystyle f(...
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Entire functions with fixed values on the integer lattice
Do there exist two distinct entire functions $f(z)$ and $g(z)$, neither of which is identically constant and such that f / g is non-constant*, such that
$$f(m + n\:i) = g(m + n\:i)$$
for all $(m, n) \...
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Interpolation from unevenly distributed points of function with compactly supported Fourier transform
It is not so hard to prove the Nyquist-Shannon theorem. It basically says a real valued function $f$ having a compactly supported Fourier transform function $\hat f$ can be interpolated precisely from ...
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What can we say about $f(x)=\prod_{n=2}^\infty(1-n^{-1/x})$?
Main Question:
What can we say about
$f(x)=\prod_{n=2}^\infty(1-n^{-1/x})$?
Is $f(x)$ integrable from $0$ to $1$? Is it continuous? If we have an affirmative answer to the question on integrability...
...
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Factorization of $\sin(\pi z)$ through Hadamard Factorization Theorem
Hadamard's Theorem states that: Let $f$ be an entire function of finite order. Denote the zeroes of $f$ by $|a_1|\leq |a_2|\leq \dots$. Then: $f$ gen$f\leq $ord$f\leq $gen$f$+1 where gen is the genus ...
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How to derive Weierstrass factorization for $\cos(z)$ from $\tan(z)=\sum_{n=0}^\infty \frac{8z}{(2n+1)^2\pi^2-4z^2}$
Given that: $\tan(z)=\sum_{n=0}^\infty \frac{8z}{(2n+1)^2\pi^2-4z^2}$
Integrate on both sides:
$-\ln(\cos(z))=-\sum_{n=0}^\infty \left( \ln\left[ (2n+1)^2-4z^2 \right]+\ln c_n\right)$
Simplify:
$\cos(...
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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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Translation symmetry by Gaussian integers for this infinite product is impossible by Liouville. What takes its place?
On $\Bbb{R}$, we can construct the convergent infinite product $$F_{\Bbb{Z}}(x) := x \prod_{n \geq 1} \prod_{|k| = n} \left(1 - \frac{x}{k} \right) = x \prod_{n \geq 1} \left(1 - \frac{x^2}{n^2} \...
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Is there a closed-form expression for $\prod_{n=1}^{\infty}(1-\frac{x}{n^3})$?
I would like to ask if for $|x|<1$, we can express the product $\prod_{n=1}^{\infty}(1-\frac{x}{n^3})$ as a function $f(x)$. I tried to use Weierstrass factorization theorem, but without much ...
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How I prove absolute convergence of this infinite product?
Why does the 1/n term needs to be 0 in order to have absolute convergence for the product? $$P=\prod_{n=1}^{\infty }u_n=\prod \frac{(n-a_1)(n-a_2)...(n-a_n)}{(n-b_1)(n-b_2)...(n-b_n)}$$
$$u_n=1-\frac{...
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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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Function on $\mathbb{C}$ with all primes as zeros?
According to the Weierstraß factorization theorem, an entire function with all primes as zeros would be (if I didn't mess up):
$$\tilde P(z) = \prod_{p \text{ is prime}} \left(1-\frac{z}{p}\right) \...
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Is this a novel factorization theorem? Solving $\sin^2(\pi x)+\sin^2\left(\frac{\pi p}{x}\right)=0$ for $x$ gives the integer factors of $p$.
I have found that
$$
\sin^2(\pi x) + \sin^2\left(\frac{\pi p}{x}\right)=0
$$
solved for $x$ defines the integer factors of $p$. Iteratively applied it also defines the prime numbers smaller than $n$ ...
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Weierstrass-type factorization (reconstruction) of a function with branch cuts
Consider a function $f(z)$ which has an infinite number of zeros (only) along the positive real axis. I will write $f(z_n) = 0$, for $z_n \in \mathbb{R}$, with $z_n \geq 0 $ and labeled by $n \in \{1,...
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Expansion of Complex Functions
I apologize for my unusual terminology, but my math training in this field is rather lacking, and not entirely in English.
Consider the set of complex functions that are holomorphic except for a ...
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Weierstrass Factorization Theorem for meromorphic functions.
On Wikipedia page about Weierstrass factorization theorem one can find a sentence which mentions a generalized version so that it should work for meromorphic functions. I mean:
We have sets of ...
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Meromorphic interpolation of number-theoretic functions
A meromorphic interpolation of the harmonic numbers is given by
\begin{align}
H_n
&= \sum_{k=1}^n \frac{1}{k} \\
&= \sum_{k=0}^{n-1} \frac{1}{k+1} \\
&= \sum_{k=0}^{n-1} \int_0^1 x^k \,\...
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Product of two holomorphic functions defined by infinite product equals to 1. [duplicate]
Let $f(z)=\prod_{n=1}^{\infty}(1+z^n)$ and $g(z)=\prod_{n=1}^{\infty}(1-z^{2n-1}).$
Let $\Omega=\{z\in \mathbb{C}, \; |z|<1\}.$
Since the series $\sum_{n=1}^{\infty}|z|^n$, and $\sum_{n=1}^{\...
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Entire function non identically zero implies that limit sequence of zeros diverges
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ an entire function and let $(a_n)_{n\geq 1} \subset \mathbb{C}^{*}$ the sequence of the zeros of $f$. Suppose that $z=0$ is a zero of $f$ of order $m\geq 1$ ...
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Show $\cosh(z) - \cos(z) = z^2\prod_{n = 1}^{\infty}\left(1 + \frac{z^4}{4\pi^4n^4} \right)$
I would like to show that $\cosh(z) - \cos(z) = z^2\prod_{n = 1}^{\infty}\left(1 + \frac{z^4}{4\pi^4n^4} \right)$.
Firstly, I have found that solutions to the equation $\cosh(z) - \cos(z) = 0$ are of ...
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Understanding infinite product of $\sin(\pi z)$
The infinite product of $\sin(\pi z)$ is said to be...
$\sin (\pi z)=\pi z\prod _{{n\neq 0}}\left(1-{\frac {z}{n}}\right)e^{{z/n}}=\pi z\prod _{{n=1}}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)$
...
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Understanding the roots in the Weierstrass factorization theorem
To construct the Weirstrass product, we first start with something of the form $$\prod_{k=1}^\infty(z-a_k)$$
The next thing we do is include exponential factors out front (which obviously don't ...
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Understanding the elementary factors in Weierstrass factorization theorem
An infinite product such as $\,\prod _{n}(z-c_{n})$ cannot converge. In order for it to converge, each factor $(z-c_{n})$ must approach 1 as $n\to \infty$. So it stands to reason that one should seek ...
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Simplifying $(\sin^6x + \cos^6x) - (\sin^4x + \cos^4x) + \sin^2x\cos^2x$
$(\sin^6x + \cos^6x) - (\sin^4x + \cos^4x) + \sin^2x\cos^2x =$
Right Answer: $0$, but I could not solve this question. Help me please.
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Construct an analytic function which has simple zeros at all $m+in$
I found a "duplicate" question: Entire function having zeros at $m+in$, and I went through some wiki pages of Weierstrass sigma function, but they don't seem to have constructions.
So basically I'm ...
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Infinite Product Expansion of Hyperbolic Functions
the following equation is from "[1970] Goodson - Distributed system simulation using infinite product expansions":
\begin{align*}
\cosh z + \left( c z+ \frac{d}{z} \right) \sinh z & = (1 + d) \...
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Weierstrass factorization theorem for non-entire functions
I know that any entire function $f(z)$ can be represented in the form of an infinite product according to the Weierstrass factorization theorem:
$$
f(z) = z^m e^{g(z)}\prod_{n=1}^\infty E_{p_n}\left(\...
user219526
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Question on infinite product of zeta function
So I’ve read Weierstrass’ factorization theorem. I sort of know how the Hadamard product of the zeta function was derived. I don’t get however, how they moved from this
$$\zeta(s)=\frac{e^{(\ln(2\pi) ...
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Analytic function $f$ on unit open disc $D$ which is not analytic on any open set $G$ which properly contains $D$
Show that there is an analytic function $f$ on unit open disc $D$ which is not
analytic on any connected open set $G$ which properly contains $D$.
My attempt : I know Weierstrass factorization ...
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Constructing a holomorphic function with a given zero set.
So here are two questions:
Construct a holomorphic and bounded function on $\mathbb{D}$ with infinitely many zeros in $\mathbb{D}$ accumulating in $i$.
And construct and entire function with only ...
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$E,F$ disjoint subsets of $D$ having no limit point in $D$ ; to show a meromorphic function with poles exactly in $E$ and zeros exactly in $F$
Let $D \subseteq \mathbb C$ be an open set and $E,F$ be disjoint subsets of $D$ having no limit point in $D$ . Then how to show that there there is a meromorphic function in $D$ such that $f$ has a ...
user228169
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$\{a_n\} \subseteq \mathbb C$ discrete set with no limit point . For any sequence $\{z_n\} $ , there is entire $f $ on $\mathbb C$ with $f(a_n)=z_n$?
Let $\{a_n\} \subseteq \mathbb C$ be a discrete set with no limit point . Then for every sequence $\{z_n\}$ of complex numbers , can we find an entire function $f:\mathbb C \to \mathbb C$ such that $f(...
user228168
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Understanding Elstrodt's proof of the $\eta$ transformation formula using the Weierstrass $\zeta$ function
In his proof of the $\eta$- transformation formula Elsrodt uses the Weierstrass $\zeta$ function. His first step in the proof is the following:
Let $\mathbb{H}$ denote the upper half-plane in $\...
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Elementary factors in the Weierstrass Factorization Theorem
Weierstrass Factorization Theorem allows representing an entire function $f$ (can be considered as an infinite polynomial) as a product involving zeros $\{a_n\}$ of $f$:
$$
f(z)=z^m e^{g(z)}\prod_{n=...
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:Duplicate: How to prove that $\cos (\pi * z) = \prod_{n \in \mathbb{Z}_{\text{odd}}} (1-\frac{2 z}{n})e^{2z/n}$ [duplicate]
Duplicate
The question was also answered here:
How can I deduce $\cos\pi z=\prod_{n=0}^{\infty}(1-4z^2/(2n+1)^2)$?
I found this formula on Wikipedia under Weierstrass factorization theorem,
$\cos (\...
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Weierstrass equation help needed
thanks in advance.
I am trying to prove that the Weierstrass equation
y^2+a1xy+a3y=x^3+a2x^2+a4x+a6 can be written in the form of:
y2=x3+ax+b ,curves are over some field K where char K≠2,3.
I found ...
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Elliptic curve, different forms of. [duplicate]
$y^2 = x^3 + mx + c$ An elliptic curve in the form defined in Wikipedia
$y^2 = x(x-A)(x+B) = x^3 +(B-A)x^2 + ABx$ Frey's curve
has no term in $x^2$, but $2$. does because from Fermat, $A=a^n$ ...
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