# 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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### 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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### 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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Given a non constant meromorphic doubly periodic function $f$ with real-independent periodes $f(z+\omega_1)=f(z+\omega_2)=f(z)$. Consider $F:=\{\lambda_1\omega_1+\lambda_2 \omega_2, \lambda_1, \... 0answers 97 views ### Derive Weierstrass product theorem from Mittag - Leffler theorem [duplicate] This question is from Ponnusamy Silvermann's Complex analysis section Mittag-Leffler Theorem and I am struck on this. Derive Weierstrass product theorem from Mittag-Leffler theorem I am confused on ... 1answer 40 views ### Proving existence of a positive sequence${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 ... 1answer 171 views ###$\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-... 0answers 51 views ### 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 ... 0answers 33 views ### 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 ... 1answer 61 views ### 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 ... 0answers 48 views ### 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, \... 1answer 70 views ### 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 ... 1answer 188 views ### 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, ... 0answers 30 views ### 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}} ... 1answer 42 views ### Efficient Meromorphic Approximation For Getting the ith Bit of a Number Thanks to this answer, I know that to get the ith 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$$However, I need this formula ... 0answers 53 views ### Prove \sum_{n=0}^{\infty} \frac{(-1)^n}{z+n} is meromorphic The question is prove that the series \sum_{n=0}^{\infty} \frac{(-1)^n}{z+n} determines a meromorphic function. So the way that I prove these kinds of questions is fix R>0 and prove that it is ... 1answer 35 views ### Is a meromorphic function determined by its boundary values? Let f: \mathbb D \to \widehat {\mathbb{C}} be a meromorphic function inside the unit disk. Assume that f is zero on the boundary and continuous in the closed disk (as a function into \widehat {\... 0answers 14 views ### Integral closure of the local ring of a riemann surface Consider the following situation: we have a holomorphic (non-constant) map X\overset{f}{\to}Y, which induces in the obvious way \mathcal{M}(X)\hookleftarrow\mathcal{M}(Y). Write also \mathcal{O}... 1answer 117 views ### If a_i\in\mathbb{R}, \omega^2+\omega+1=0, and \sum_{i=1}^n\frac{1}{a_i+\omega^k} =2\omega^{2k} for k=1,2, find \sum_{i=1}^n\frac{1}{a_i+1}. In this question, \omega is the complex cube root of 1 and a_i \in \mathbb R. If$$\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\,,$$... 1answer 43 views ### Meromorphic continuation of 1+z+z^2+z^3+\ldots Consider the function f(z)=1+z+z^2+z^3+\ldots. This series is absolutely convergent on the disc |z|<1 and is equal to 1/(1-z) in this region. Now, 1/(1-z) is a meromorphic function on \... 1answer 52 views ### Meromorphic function with a removable singularity and a few poles Here is the question: Let f be a meromorhic function on \mathbb{C}, having poles at the following three points: z=5, z=1+3i and z=3-4i. Also, let f have one removable singularity at z=3.... 2answers 84 views ### Basis of L(4p_0) for a torus and induced embedding into \mathbb{P}^3 I am trying to solve the following exercise; in more general situations also I do not know how to address the problem of finding the equations for the image of a Riemann surface under the embedding ... 2answers 68 views ### Show that a meromorphic continuation exists I am preparing for the complex analysis qualifying exam, and I recently came across this problem: Show that$$F(z)=\int_1^\infty\frac{t^z}{\sqrt{1+t^3}}\,dt$$defines an analytic function on \{z\in\... 1answer 33 views ### 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 + ... 0answers 59 views ### 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 ... 2answers 66 views ### 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 ... 1answer 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 ... 1answer 38 views ### 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 ... 1answer 115 views ### 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:... 1answer 59 views ### 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? ... 4answers 95 views ### 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})}{... 0answers 29 views ### 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 ... 0answers 49 views ### 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 ... 1answer 69 views ### 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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### 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 ...