Questions tagged [meromorphic-functions]

Meromorphic functions are complex-valued functions which are holomorphic everywhere on an open domain except a set of isolated points which are poles. Consider also using the (complex-analysis) tag.

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Triangulating Riemann surfaces by using non-constant meromorphic functions.

Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result: Theorem (...
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If $f$ is entire with $f\not\equiv 0$, then $f'/f$ is meromorphic

Suppose $f$ an entire function with $f'(z)\ne0$ for all $z\in\Bbb C$. How can I see that $f'/f$ is meromorphic? I know that if $f$ and $g$ are holomorphic with $g\not\equiv0$, then $f/g$ is ...
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Defining meromorphic functions and forms on a Riemann surface using a single formula in a single chart

I am working out of Miranda's Algebraic Curves and Riemann Surfaces. In the book, we define differential forms for a Riemann surface to be a collection of forms satisfying a compatibility condition {$\...
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Rouché's Theorem in complex analysis on the relation of the number of zeros and poles of meromorphic functions in a region

This question is from my son referenced in my earlier question, https://mathoverflow.net/questions/382003/need-advice-or-assistance-for-son-who-is-in-prison-his-interest-is-scattering-t/382105?...
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Cauchy-Riemann equations: Meromorphic Function

A meromorphic function is a function that is holomorphic on all domain except for a set of isolated points. I know that a holomorphic function, by definition, satisfies the Cauchy-Riemann equations ...
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Definition of Series of Meromorphic Functions

Let $D\subset \Bbb{C}$ be non-empty, open, and $\{f_n\}_{n=1}^{\infty}$ a sequence of meromorphic functions on $D$. In Henri Cartan's complex analysis book, the following definition is given: We say ...
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the nessesary condition of branch of logarithm

Let $U$ be an open subset of $\mathbb{C}$.Let $z_0$ be a point in U,and suppose that f is a meromorphic function on U with a pole at $z_0$.Prove that there is no holomorphic function g:U$\setminus${$...
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Meromorphic matrix

I am starting to study some documents for my thesis, maybe this question is basic, but I don´t know precisely what is. What is a meromorphic matrix?. Where i can read about of the precisely ...
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What does meromorphic continuation mean?

Sorry if this question is slightly out of context. I've learned that $\zeta(k)=\sum\limits^{\infty}_{z=1} \frac{1}{z^k}$ has meromorphic continuation to $Re(s)>0$. I know that, from convergence ...
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If $f$ is holomorphic on $U\setminus D$ and $D$ consists of poles isolated singularities, is there any isolated singularity outside $D$?

Let $X$ be a normed vector space. 1.: If $\Omega\subseteq\mathbb C$ is open and $f:\Omega\to X$ is holomorphic, then $z_0\in\mathbb C\setminus\Omega$ is called an isolated singularity of $f$ if there ...
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Decomposition of a meromorphic function into a singular and holomorphic part

Let $X$ be a normed space, $\Omega\subseteq\mathbb C$ be open and $f:\Omega\to X$ be meromorphic in the sense that for all $z_0\in\Omega$, there is a $k_{z_0}\in\mathbb N_0$, $\left(a^{(z_0)}_k\right)...
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If a function is meromorphic on a ball, the set of poles is finite

Let $X$ be a normed space, $\Omega\subseteq\mathbb C$ be open and $f:\Omega\to X$ be meromorphic in the sense that for all $z_0\in\Omega$, there is a $k_{z_0}\in\mathbb N_0$, $\left(a^{(z_0)}_k\right)...
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Laurent series expansion of holomorphic/meromorphic functions

I'm not familiar with complex analysis, but I need to catch up a basic fact about holomorphic and meromorphic functions. Let $\Omega\subseteq\mathbb C$, $f:\Omega\to\mathbb C$ and $z_0\in\Omega$. Am ...
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Suppose that f is meromorphic in $\mathbb{C}$ such that $|f(z)|< |z|^{k}, \ |z|> R.$ Show that f is rational.

Suppose that f is meromorphic in $\mathbb{C}$ and that there are $ K \in \mathbb{N}$ e positive numbers C,K such that $|f(z)|< |z|^{k}, \ |z|> R.$ Show that f is rational. To make an argument ...
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Example of a non-normal family of meromorphic functions defined on the unit disk whose derivative is normal.

It is well known that if a family of meromorphic functions is not normal ( a family of meromorphic functions is said to be normal if each sequence of functions in the family has a subsequence which ...
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Transcendence of limit function under uniform convergence.

Suppose $\{f_n\}$ is a sequence of non-constant meromorphic functions defined on some domain $D\subset\mathbb{C}.$ Assume that $\{f_n\}$ converges uniformly (w.r.t spherical metric) on $D$ to some non-...
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Identity with moduli for a holomorphic function on a disc [duplicate]

Let $f(z) = \sum a_n z^n$ be a holomorphic function on the unit open disc which takes it to itself. How can I show that for any real $0 <r <1$ it holds that $\sum |a_n|^2 r^{2n} = \frac{1}{2 \pi}...
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Global Injectivity Criterion of Meromorphic Functions

Suppose we have a locally injective meromorphic function $f$ on some domain $D$. The three interesting cases for $D$ are the plane, the disc and an arbitrary domain. Suppose its poles are $P = \...
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help with meromorphic doubly periodic function

Given a non constant meromorphic doubly periodic function $f$ with real-independent periodes $f(z+\omega_1)=f(z+\omega_2)=f(z)$. Consider $F:=\{\lambda_1\omega_1+\lambda_2 \omega_2, \lambda_1, \...
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Derive Weierstrass product theorem from Mittag - Leffler theorem [duplicate]

This question is from Ponnusamy Silvermann's Complex analysis section Mittag-Leffler Theorem and I am struck on this. Derive Weierstrass product theorem from Mittag-Leffler theorem I am confused on ...
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Proving existence of a positive sequence ${a_n}$ such that $f(z)= \sum_{n=1}^{\infty} a_n /( z- b_n )$ is analytic except $z =b_n$

This question was asked in a masters exam for which I am preparing and I am not able to solve it. Suppose that $0< |b_1| \leq |b_2| \leq ...(b_n \to \infty).$ Show that there exists a positive ...
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$\prod_{n=1}^{\infty} (1- z/ a_n) $ is entire iff $\sum_{n=1}^{\infty} 1/(z-a_n) $ is meromorphic

This question was asked in a masters exam previous year paper and I was unable to prove it. Show that the $$\prod_{n=1}^{\infty} (1- \frac{z}{a_{n}}) $$ is entire iff $$\sum_{n=1}^{\infty} \frac{1}{z-...
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Construct an analytic function f in |z|<R such that f has zeroes only at z=-R +1/n $n \in \mathbb{N}$

This question is from Ponnusamy and Silvermann's Complex variables with applications ( Chapter : Entire and meromorphic functions). Construct an analytic function f in |z|<R such that f has zeroes ...
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Noether folrmula, Riemann Roch , Reimann Surfaces, Global holomorphic sections of a Line bundle

Let $K$ be the canonical line bundle of a Riemann Surface $M$ of genus $g.$ Consider the pullback of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the global ...
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$f$ meromorphic prove that $f$ is not constant

I'm struggling with the following problem. Any help would be a really welcomed. Let $f$ a meromorphic function on $\mathbb C$ s.t $g(w)= f(\frac{1}{w})$ is meromorphic for $w=0$. We say that $f$ is ...
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Definition of an algebraic curve

I'm reading the book of Miranda (Algebraic Curves and Riemann Surfaces), and in the text he give the following definition: Let $X$ be a compact Riemann surface, then $X$ is a algebraic curve if, $\...
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1answer
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Meromorphic differentials and the pullback.

In Diamond & Shurman's A First Course in Modular Forms Section 3.3, the authors naively treat so-called "meromorphic differentials". It seems that the spaces they denote $\Omega^{\otimes ...
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Doubly periodic meromorphic function with prescribed poles and zeros

The field of the meromorphic functions on a complex torus $\mathbb{C} \mathbin{/} \Lambda$ is $\mathbb{C}(\wp, \wp')$, where $\wp$ is the weierstrass p-function to the lattice $\Lambda$. Furthermore, ...
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Determining a meromorphic function by its poles.

Given a series of complex numbers $\{a_j\}_{j\in\mathbb{Z}}$, there exists a meromorphic function having simple poles only at all $\mathbb{Z}$ with the residues equal to $\{a_j\}_{j\in\mathbb{Z}}$ ...
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Efficient Meromorphic Approximation For Getting the ith Bit of a Number

Thanks to this answer, I know that to get the $i$th bit of a number $n$, you can do $$\left\lfloor\frac{n}{2^i}\right\rfloor-2\left\lfloor\frac{n}{2^{i+1}}\right\rfloor$$ However, I need this formula ...
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Prove $\sum_{n=0}^{\infty} \frac{(-1)^n}{z+n}$ is meromorphic

The question is prove that the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{z+n}$ determines a meromorphic function. So the way that I prove these kinds of questions is fix $R>0$ and prove that it is ...
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Is a meromorphic function determined by its boundary values?

Let $f: \mathbb D \to \widehat {\mathbb{C}}$ be a meromorphic function inside the unit disk. Assume that $f$ is zero on the boundary and continuous in the closed disk (as a function into $\widehat {\...
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Integral closure of the local ring of a riemann surface

Consider the following situation: we have a holomorphic (non-constant) map $X\overset{f}{\to}Y$, which induces in the obvious way $\mathcal{M}(X)\hookleftarrow\mathcal{M}(Y)$. Write also $\mathcal{O}...
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If $a_i\in\mathbb{R}$, $\omega^2+\omega+1=0$, and $\sum_{i=1}^n\frac{1}{a_i+\omega^k} =2\omega^{2k}$ for $k=1,2$, find $\sum_{i=1}^n\frac{1}{a_i+1}$.

In this question, $\omega$ is the complex cube root of $1$ and $a_i \in \mathbb R$. If $$\sum_{i=1}^n \frac{1}{a_i + \omega} =2\omega ^2$$ and $$\sum_{i=1}^n \frac{1}{a_i + \omega ^2} =2\omega\,,$$ ...
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Meromorphic continuation of $1+z+z^2+z^3+\ldots$

Consider the function $f(z)=1+z+z^2+z^3+\ldots$. This series is absolutely convergent on the disc $|z|<1$ and is equal to $1/(1-z)$ in this region. Now, $1/(1-z)$ is a meromorphic function on $\...
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Meromorphic function with a removable singularity and a few poles

Here is the question: Let $f$ be a meromorhic function on $\mathbb{C}$, having poles at the following three points: $z=5$, $z=1+3i$ and $z=3-4i$. Also, let $f$ have one removable singularity at $z=3$....
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Basis of $L(4p_0)$ for a torus and induced embedding into $\mathbb{P}^3$

I am trying to solve the following exercise; in more general situations also I do not know how to address the problem of finding the equations for the image of a Riemann surface under the embedding ...
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Show that a meromorphic continuation exists

I am preparing for the complex analysis qualifying exam, and I recently came across this problem: Show that $$F(z)=\int_1^\infty\frac{t^z}{\sqrt{1+t^3}}\,dt$$ defines an analytic function on $\{z\in\...
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Existence of an algebraic function on a disc, in an elementary way

The meromorphic functions on an open disc $\Delta$ in $\mathbf{C}$ form a field $M(\Delta)$. How to show in as elementary a way as possible that for every polynomial $P(X) = X^n + a_1 X^{n-1} +\dots + ...
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A good definition for “very ample divisor” on Riemann surface

I'm studying Riemann surfaces, I have no experience of other algebraic geometry and of line bundles in particular. A divisor was defined to me as an element of the free abelian group on a RS. I'm ...
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Space of meromorphic functions is not finitely generated

During a lesson of the course on Riemann Surfaces our lecturer made the following remark, saying this could be proved as an exercise: The space $\mathcal{M}(X)$ of meromorphic functions on a compact ...
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161 views

Order of the sum of two meromorphic functions

I haven't been around complex analysis at some time now and I need it for Riemann surfaces and I a bit confused about something. I know that if I have a meromorphic function $f$ with order $k$ at $p$ ...
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Behaviour of meromorphic functions near poles.

Let $F(z)$ be a meromorphic function on a domain $\Omega$ with a pole at $z=a .$ Let $C \subset \Omega$ be a simple closed contour with the point $z=a$ enclosed in its interior domain $I,$ and assume ...
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Lifting of meromorphic function along a finite morphism

I am currently reading the book "Geometry of algebraic curves II", by Arbarello, Cornalba and Griffiths, and I am having some difficulties understanding a passage p.105. The setting is the following:...
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Does this strategy of characterizing poles always work?

I stumbled upon a fast way to characterize poles of order $m$ of a meromorphic function $f$ (on some open set $\Omega$) in this answer here. My question is, does this general strategy always work? ...
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Is $\frac{\cos(\frac{1}{z})}{z^2}$ meromorphic or Not?

my professor used the Cauchy Residue Theorem to evaluate the path integral (along the positively-oriented unit circle about the origin with winding number 3) $$\int_{\gamma}\frac{\cos(\frac{1}{z})}{...
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If f is a meromorphic function then f(z+c) is also so of the same order

I am stuck with the statement given in the title. If $f$ is a meromorphic function, and $c\neq 0$, a complex constant, then $f(z+c)$ is also a meromorphic function of the same order. If $f$ is a ...
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Functions that separate points and tangents on an Algebraic Curve

I am trying to do an exercise from Rick Miranda's Book that goes like this Let $X$ be an algebraic Curve, show using the compactness of $X$ that there are a finite number of global meromorphic ...
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Transcendence degree of $\mathcal{M}(x)$ in $\mathbb{C}$, and $\mathcal{M}(X)$ is finitely generated over $\mathbb{C}$

I have been reading about Riemann surfaces from Rick Miranda's Book and now the term transcendent degree has come in to play, but I have no knowledge of field theory so I dont quite understand it I ...
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1answer
67 views

Meromorphic functions of non integral order

We know that an entire function of non integral order has infinitely many zeros. What can be said about the zeros and poles of a meromorphic function of non integral order? In fact, is there any ...