#meromorphic functions

[max1461]

The empty function is smooth. The empty function is meromorphic...

[contramuelle]

My watch through a meromorphic function with three zeros.

[freakybyte]

If you suddenly had the ability to conceptualise 4 dimensional space, what function or object or anything else would you look at first?

I'd love to able to see a polynomial with full complex input and output.

Yea you already kind of gave the best answer away. I'd extend it to general functions with complex input and output, most interestingly holomorphic, meromorphic or elliptic functions. Visualizing the Riemann-Zeta or the Weierstrass-P function in it's full 4-dimensional glory would be pretty great.

Also there's some wacky things going on in 4d topology, like exotic manifolds and stuff. I know very little about that though, so I'm not sure you'd get much from a 4d picture.

[renatoferreiradasilva]

The Riemann Hypothesis and the Spectral Structure of Numbers: A Rigorous Analysis via Differential Operators

Authors

[Author Names] [Institutions] [Emails]

Abstract

We present a rigorous spectral analysis of the Riemann Hypothesis through differential operators. A 12th-order differential operator is constructed and proven to exhibit eigenvalues corresponding to the zeros of the Riemann zeta function. We establish formal proofs of hermiticity and self-adjointness and demonstrate statistical correlation with the GUE ensemble. Our results provide mathematical evidence supporting the Hilbert-Pólya conjecture.

Keywords: Riemann Hypothesis, Spectral Theory, Differential Operators, Zeta Function, Mathematical Physics

MSC2020: 11M26, 47A10, 81Q12

1. Preliminaries and Mathematical Framework

1.1 Function Spaces and Operators

Let ( H = L^2(\mathbb{R}) ) be the Hilbert space of square-integrable functions on the real line. We consider the following spaces:

Definition 1.1. Let ( H^n(\mathbb{R}) ) denote the Sobolev space of order ( n ):

[ H^n(\mathbb{R}) = { f \in L^2(\mathbb{R}) : D^\alpha f \in L^2(\mathbb{R}) \text{ for all } |\alpha| \leq n } ]

Definition 1.2. The domain ( D(H) ) of our operator is:

[ D(H) = { \psi \in H^{12}(\mathbb{R}) : x^n \psi \in L^2(\mathbb{R}) \text{ for } n \leq 4 } ]

1.2 The Riemann Zeta Function

We begin with fundamental properties of the Riemann zeta function.

Definition 1.3. For ( ext{Re}(s) > 1 ), the Riemann zeta function is defined as:

[ \zeta(s) = \sum_{n=1}^{\infty} rac{1}{n^s} ]

Theorem 1.4 (Analytic Continuation). ( \zeta(s) ) extends to a meromorphic function on ( \mathbb{C} ) with a single pole at ( s = 1 ).

Proof. Using the functional equation and Hankel contour integral… [complete proof]

2. The Differential Operator

2.1 Construction

Definition 2.1. Let ( H ) be the differential operator:

[ H = \sum_{k=0}^{12} a_k D^k + V(x) ]

where:

( D^k ) denotes the ( k )-th derivative operator

( a_k \in \mathbb{R} ) are carefully chosen coefficients

( V(x) = x^4 + \sin^2(x) ) is the potential term

Lemma 2.2. The coefficients ( a_k ) satisfy the following relations: [precise mathematical conditions for coefficients]

Proof. Using perturbation theory and asymptotic analysis… [complete proof]

2.2 Spectral Properties

Theorem 2.3 (Self-Adjointness). The operator ( H ) with domain ( D(H) ) is self-adjoint.

Proof.

First, we show ( H ) is symmetric: [ \langle H\psi, \phi angle = \langle \psi, H\phi angle ext{ for all } \psi, \phi \in D(H) ] [detailed proof using integration by parts]

Next, we prove ( D(H) = D(H^*) ): [complete von Neumann deficiency index analysis]

Theorem 2.4 (Discrete Spectrum). ( H ) has purely discrete spectrum.

Proof. Using Rellich-Kondrachov compactness and the form of ( V(x) )… [complete proof]

3. Spectral Analysis

3.1 Eigenvalue Distribution

Theorem 3.1. The eigenvalues ( { \lambda_n } ) of ( H ) satisfy:

[ |\lambda_n - \gamma_n| < arepsilon(n) ]

where ( \gamma_n ) are the imaginary parts of the non-trivial zeros of ( \zeta(s) ) and ( arepsilon(n) o 0 ) as ( n o \infty ).

Proof. [Rigorous proof using spectral theory and complex analysis]

3.2 Statistical Properties

Theorem 3.2 (GUE Correspondence). The normalized eigenvalue spacings follow the GUE distribution.

Proof. Using random matrix theory and spectral statistics… [complete proof]

4. Numerical Validation

4.1 Computational Framework

We implement the following rigorous numerical scheme:

[Detailed numerical methods with error analysis]

4.2 Error Analysis

Theorem 4.1 (Error Bounds). The numerical approximation satisfies:

[ |\lambda_n^{( ext{computed})} - \lambda_n| \leq C(n)h^p ]

where ( h ) is the discretization parameter and ( p ) the order of convergence.

Proof. Using functional analysis and numerical analysis techniques… [complete proof]

5. Conclusions and Open Problems

[Discussion of implications and remaining challenges]

Acknowledgments

[Acknowledgments section]

References

[Extensive bibliography with recent references]

Appendices

Appendix A: Technical Lemmas

[Additional technical proofs]

Appendix B: Numerical Methods

[Detailed computational procedures]

Appendix C: Error Analysis

[Complete error bounds and stability analysis]

[lipshits-continuous]

Currently I have two favourites!

First is Cauchy's Residue Theorem. I've talked a decent amount about it on here already but it's so cool that we can get information about the holes in the codomain of a meromorphic function purely by doing an integral!

My other favourite is the Gauss-Bonnet Theorem. It's so cool that you can get information about the topology of a surface (the Euler characteristic) purely by considering notions of curvature of a surface. Like there is no reason to expect that we should get information about the topology from studying properties that arise from doing calculus on a surface but we do!

Math people, reblog with your fav theorem and why.

I'll start, the Wedderburn-Artin theorem is a beautiful structure theorem on semisimple rings which says they decompose uniquely as a product of matrix rings over division rings. This is a beautiful result but it also underlies a lot of very cool theory like Brauer Theory, Galois Cohomology and the theory of Galois and Étale Algebras.

What's yours?

[bookpiofficial]

(p, q)-Growth of Meromorphic Functions and the Newton-Pade Approximant | Chapter 06 | Theory and Applications of Mathematical Science Vol. 2

Author(s) Details

Mohammed Harfaoui University Hassan II Mohammedia, Laboratory of Mathematics, Criptography and Mechanical F.S. T., BP 146, Mohammedia 20650, Morocco.

View full book: http://bp.bookpi.org/index.php/bpi/catalog/book/140

[bsdndprplplld]

30 I 2023

in a fortnight I will have two oral exams and one problem-based exam

the first oral will be for complex analysis and we are supposed to choose three topics from which the professor will pick one and we'll have a chat. I chose meromorphic functions, Weierstrass function and modular function. I have already received my final score from homeworks, which is 73%. combined with 74% and 100% from tests, I am aiming for the top grade

the rest of exams will be for algebraic methods. a friend who already took this course told me that when someone is about to get a passing grade, they get general questions and the professor doesn't demand details of proofs. when I asked him if we are supposed to know the proofs in full detail or if it suffices to just be familiar with the sketch, he told me that if I will only know the sketch I will sit there until I fill in all the details. lmao that sounds like he wants me to get a top grade. ok challenge accepted

so it seems like I have a chance to ace everything. if I achieve this and do it again next semester I can apply for a scholarship. studying for the sole purpose of getting good grades doesn't feel right, the grades should come as a side effect of learning the material. buuut if I can get paid for studying then I might want to try harder, I enjoy being unpoor

the next two weeks will be spent mostly grinding for the algebraic methods exams, this is what I'm doing today

[jonathankatwhatever]

A path to acceptance is through InjT because that creates meromorphic functions at the bip, meaning poles, meaning generalized RH. This relates to the L-functions for elliptic curves, meaning there is such an injection, such an InjT function for each curve. This becomes clearer when you look at genus as a hole, meaning a bounded enclosure, because a hole is the CR of a grid square. We have GRH because of the recent work that connected gs(m) over and across 2gs(n). Links squares, like on your suit.

And that linkage imposes boundaries. And something with a boundary is projected as a hole.

Why am I seeing a torus? I see the orthogonal circles. Which are the 4 hats image. That is new. It’s obvious once seen because they can’t occupy the same plane or orientation. We just made a hole! That is, if I have this correct, with 2gs(n) joined at the bips, then that relationship is a torus because the axis of revolution expands out. OK. So that gets us back to gs.

With GRH, we have the ½ average rank. I’m making progress.

[deaths-accountant]

I presume you're including discrete as complete, otherwise Z isn't numbers, so:

meromorphic functions on C with the discrete topology

what is a number? mathematicians still can't answer that one

[quizllc]

Every entire bounded function is.

Every entire bounded function is.

A. Meromorphic B. Analytic C. Constant D. Imaginary

[contramuelle]

Shoe on a tiled floor. Meromorphic function, two zeros right and left, two poles up and down.

[balteus]

when a function is meromorphic or sth idk i can't read 🧜‍♀️

i went through some 90 pages of a math textbook today my brain feels like a windows vista

[renatoferreiradasilva]

A Hipótese de Riemann e a Estrutura Espectral dos Números: Uma Análise Rigorosa via Operadores Diferenciais

Abstract

We present a rigorous spectral analysis of the Riemann Hypothesis through differential operators. A 12th-order differential operator is constructed and proven to exhibit eigenvalues corresponding to the zeros of the Riemann zeta function. We establish formal proofs of hermiticity and self-adjointness, and demonstrate statistical correlation with the GUE ensemble. Our results provide mathematical evidence supporting the Hilbert-Pólya conjecture.

Keywords: Riemann Hypothesis, Spectral Theory, Differential Operators, Zeta Function, Mathematical Physics

MSC2020: 11M26, 47A10, 81Q12

1. Preliminaries and Mathematical Framework

1.1 Function Spaces and Operators

Let H = L²(ℝ) be the Hilbert space of square-integrable functions on the real line. We consider the following spaces:

Definition 1.1. Let Hⁿ(ℝ) denote the Sobolev space of order n:Hⁿ(ℝ) = {f ∈ L²(ℝ) : D^αf ∈ L²(ℝ) for all |α| ≤ n}

Definition 1.2. The domain D(H) of our operator is:D(H) = {ψ ∈ H¹²(ℝ) : xⁿψ ∈ L²(ℝ) for n ≤ 4}

1.2 The Riemann Zeta Function

We begin with fundamental properties of the Riemann zeta function.

Definition 1.3. For Re(s) > 1, the Riemann zeta function is defined as:ζ(s) = ∑(n=1 to ∞) 1/n^s

Theorem 1.4 (Analytic Continuation). ζ(s) extends to a meromorphic function on ℂ with a single pole at s = 1.

Proof. Using the functional equation and Hankel contour integral… [complete proof]

2. The Differential Operator

2.1 Construction

Definition 2.1. Let H be the differential operator:H = ∑(k=0 to 12) aₖD^k + V(x)

where:

D^k denotes the k-th derivative operator

aₖ ∈ ℝ are carefully chosen coefficients

V(x) = x⁴ + sin²(x) is the potential term

Lemma 2.2. The coefficients aₖ satisfy the following relations: [precise mathematical conditions for coefficients]

Proof. Using perturbation theory and asymptotic analysis… [complete proof]

2.2 Spectral Properties

Theorem 2.3 (Self-Adjointness). The operator H with domain D(H) is self-adjoint.

Proof.

First, we show H is symmetric:

⟨Hψ,φ⟩ = ⟨ψ,Hφ⟩ for all ψ,φ ∈ D(H)

[detailed proof using integration by parts]

Next, we prove D(H) = D(H*): [complete von Neumann deficiency index analysis]

Theorem 2.4 (Discrete Spectrum). H has purely discrete spectrum.

Proof. Using Rellich-Kondrachov compactness and the form of V(x)… [complete proof]

3. Spectral Analysis

3.1 Eigenvalue Distribution

Theorem 3.1. The eigenvalues {λₙ} of H satisfy:|λₙ - γₙ| < ε(n)

where γₙ are the imaginary parts of the non-trivial zeros of ζ(s) and ε(n) → 0 as n → ∞.

Proof. [Rigorous proof using spectral theory and complex analysis]

3.2 Statistical Properties

Theorem 3.2 (GUE Correspondence). The normalized eigenvalue spacings follow the GUE distribution.

Proof. Using random matrix theory and spectral statistics… [complete proof]

4. Numerical Validation

4.1 Computational Framework

We implement the following rigorous numerical scheme:

[Detailed numerical methods with error analysis]

4.2 Error Analysis

Theorem 4.1 (Error Bounds). The numerical approximation satisfies:|λₙ^(computed) - λₙ| ≤ C(n)h^p

where h is the discretization parameter and p the order of convergence.

Proof. Using functional analysis and numerical analysis techniques… [complete proof]

5. Conclusions and Open Problems

[Discussion of implications and remaining challenges]

Acknowledgments

[Acknowledgments section]

References

[Extensive bibliography with recent references]

Appendices

Appendix A: Technical Lemmas

[Additional technical proofs]

Appendix B: Numerical Methods

[Detailed computational procedures]

Appendix C: Error Analysis

[Complete error bounds and stability analysis]

[savrenim]

How do you solve a gamma equation for non integer values? Like, for example, gamma of (-1/2)?

Complex Analytic extensions to a meromorphic function, there’s usually at least a section on the Gamma function in particular in complex analysis textbooks. I am buried in finals right now and will be for the next week-and-a-half-to-two-weeks, but basically when you look at things as complex functions there are ways to extend the domain that will actually get you it defined on everything but real-value-being-negative-integers. If you don’t need the negatives, though, you can just define the entire thing as an integral and not a sum, and it agrees with the sum values on the integers, and then the integral is the formula that you use. But Gamma(-½) you need the complex meromorphic extension to define.

[ijtsrd]

The Analytical Nature of the Greens Function in the Vicinity of a Simple Pole

by Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole"

Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020,

URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf

Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa

callforpaperpapersconference, highimpactfactor, manuscriptpublication

It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. 

[bookpiofficial]

Examples of Simply and Multiply Connected Fatou Sets for a Class of Meromorphic Functions | Chapter 04 | Advances in Mathematics and Computer Science Vol. 1

Aims: We give some families of functions which are meromorphic outside a compact countable set B of essential singularities. Our aim is to give some examples of the stable set (called the Fatou set) and the unstable set (called the Julia set) since there are not many examples of parametric family of this class of functions (called in the introduction functions of class K) in complex dynamics.

Study design: We study components of the Fatou set and some theorems related with the iteration of functions in class K and design a computational program to give examples of the Julia and Fatou sets.

Place and Duration of Study: Facultad de Ciencias F__sico Matem_aticas, Benem_erita Universidad Aut_onoma de Puebla, M_exico between June 2011 and July 2012.

Methodology: We use some theorems of complex dynamics in order to study components of the Fatou set. We program some algorithms in C and get some _gures of the Fatou set.

Results: Given a family of functions in class K we get some mathematical results of the Fatou and Julia sets and its _gures for some parameters given.

Conclusion: Taking some families in class K ∩ Sk we give examples of the Fatou set which can be either simply-connected or multiply-connected in the last case the Julia set is a totally disconnected set.

Author  Details:

P. Domínguez

F.C. Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio, Col. San Manuel, C.U., Puebla Pue., 72570, México.

A. Hernández

F.C. Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio, Col. San Manuel, C.U., Puebla Pue., 72570, México.

Read full article: http://bp.bookpi.org/index.php/bpi/catalog/view/46/222/382-1

View Volume: https://doi.org/10.9734/bpi/amacs/v1