Questions tagged [weierstrass-factorization]

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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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1answer
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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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131 views

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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403 views

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(\...
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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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64 views

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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1answer
99 views

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 ...
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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(...
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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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637 views

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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176 views

: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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Detailed explanation of the Γ reflection formula understandable by an AP Calculus student

In my recent question about the Fransén-Robinson constant, an answer was given using the Gamma reflection formula. However, as an AP Calculus student, I didn't quite understand how the reflection ...
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1answer
44 views

When is $a(z) = b(c(z)) $?

Let $a(z)$ be a given transcendental entire function. When is $a(z)=b(c(z))$ where $b,c$ are also transcendental entire functions ? How to find such $b,c$ ? In particular when $a$ is given by a ...
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Weierstrass Product of $z\cos(z)-\sin(z)$

On a black board in the floor of my university was this question as a puzzle, and I would like to know why the $\exp(g(z))$ in the Weierstrass Product is constant $1$, in the question one answer ...
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4answers
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Factorization of a polynomials in complex number.

Factorize this expression: $$a^2+b^2+c^2-ab-bc-ca.$$ The result is $$(a+b\Omega+c\Omega^2)(a+b\Omega^2+c\Omega)$$ How I can get $\Omega$ here?What's the approach?
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Weierstrass factorization theorem and primality function

I'm interested in application of the Weierstrass factorization theorem to the primality function. Let $np(x)\colon \mathbb N\to \mathbb N$ is a "not-prime" function: $$ np(x) = \begin{cases}1, & \...