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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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 there exists a function f which is meromorphic on C and satisfies |f(z)|≥|z| at all those points where f is holomorphic? [closed]

Is there exists a function $f$ which is meromorphic on $\mathbb{C}$ and satisfies $|f(z)|\geq |z|$ at all those points where $f$ is holomorphic? Is such a function is unique and entire?
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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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41 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 $z=3$....
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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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44 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\...
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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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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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Nevanlinna characteristic functions for constant function

What can we say about the Nevanlinna characteristic functions for the constant function $f(z)\equiv 0$. Clearly, $m(r,f)=0$. But, for a small $r\neq 0$, we will have $n(r,f)=\infty$. So, how can we ...
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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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Transcendental meromorphic functions

What is the exact definition of a transcendental meromorphic function? I am unable to find any exact definition to it. Please help.
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The morphisms induced by two Cartier divisors

Let X be a projective variety. We consider two Cartier divisors $D,E$ globally generated, i.e base point free, such that $E\geq D$. We have two morphisms $\phi_D: X\to \mathbb{P}(H^0(X, O_X(D))^*)$ ...
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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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Meromorphic functions are either transcendental or rational

As the title suggests, I am trying to wholly prove that meromorphic functions are either transcendental or rational. I feel that it is obvious by the definition but am unable to start. Can anyone ...
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Please help me find the zeros of the following function

Let $g(z)=e^{z^2}-\frac{z}{e^{1+2z}-1}$. Clearly, $g\not\equiv 0$ since the function $\frac{z}{e^{1+2z}-1}$ is meromorphic and also $e^{z^2}$ is entire. But, what can be said about the zeros of $g(z)$?...
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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 ...
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Show that $\dfrac{\pi}{\sin \pi z} = \dfrac{1}{z} + \sum_{n \in \mathbb N} (-1)^n \left(\dfrac{1}{z-n} + \dfrac{1}{z+n}\right)$

Show that: $$ \dfrac{\pi}{\sin \pi z} = \dfrac{1}{z} + \sum_{n \in \mathbb N} (-1)^n \left(\dfrac{1}{z-n} + \dfrac{1}{z+n}\right) $$ where $z\in \mathbb C \backslash \mathbb Z$ Let's note $\phi(z) =...
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Convergence exponent and order of meromorphic functions

Can anyone tell me the relationship between the convergence exponent of the zeros (and poles) of a meromorphic function $f$. I have seen that for an entire function, the convergence exponent of the ...
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49 views

Finite Number of Global Meromorphic Functions Separating Points and Tangents on a Compact Riemann Surface

In Miranda's "Algebraic Curves and Riemann Surfaces," there is a problem that asks the reader the show, using the compactness of a Riemann Surface $X$, that there are only a finite number of global ...
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89 views

Suppose that an entire function $f$ is holomorphic at infinity. Show that $f$ is constant.

We say that that a function $f$ is meromorphic at infinity if the function $g(z) = f(1/z)$ has a pole at $0$. Suppose that an entire function $f$ is holomorphic at infinity. Show that $f$ is constant....
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Compute the integral $\int_0^\infty \frac{\sin(x)}{x}dx$

Compute the integral $$\int_0^\infty \frac{\sin(x)}{x}dx$$ via complex integration of the meromorphic function $f(z) = e^{iz}/z$. My attempt: $\int_0^\infty \frac{\sin(x)}{x} dx = \lim_{R \to\infty} ...
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92 views

Study of a complex function (2)

Consider the function $f(z)=(4z^2+1)\tanh (\pi z)$. Determine its singularities, and, in particular, the residues in the poles. This first part should be easy: the singularities of $\tanh (\pi z)$ ...
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Meromorphic functions on the Riemann sphere , generalization of a result

I had an exercise that goes like this that i was able to solve but was wondering if it was true for more Riemann surfaces , hopefully something not isomorphic to $\mathbb{C}_{\infty}$. Let $f$ and $...
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Uniform convergence of a series of complex functions

Let $$f(z)=\sum_{n=0}^\infty \frac 1 {z+n^2}\ .$$ Prove that $f$ is holomorphic in $\mathbb C \setminus \{ -n^2:n\in \mathbb N\} $, and determine the type of the singularities in $\{ -n^2:n\in \mathbb ...
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Showing that the difference of these 2 meromorphic functions are bounded

I have the functions $g(z)=\frac{\pi^2}{sin^2(\pi z)}$ and $f(z)=\sum_{n=0}^\infty\frac{1}{(z+n)^2}+\sum_{n=1}^\infty\frac{1}{(z-n)^2}$. I need to show that the function $f-g$ is bounded and ...
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Meromorphic function is holomorphic on a compact Riemann Surface except for a given point $p$.

I encounter the following problem. Given a compact Riemann surface $X$, and for any point $p \in X$, prove there exists a meromorphic function $f$ such that $f$ is holomorphic on $X\setminus \{p\}$. ...
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correspondence between meromorphic functions on complex torus vs. $\mathbb{C}$ itself

I have an integration to do along the path which is the border of a parallelogram, namely, I need to show using integration the sum of the orders on the complex torus is $0$. I reparameterized it ...
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40 views

A series expansion formula for the residues of a function having an infinite number of poles

Question : Let $f(z)$ be such that along the path $C_N$ (illustrated), $|f(z)| \leq M/{|z|^k}$, where $k > 1$, and $M$ are constants independent of $N$. Show that $$\sum_{n=-\infty}^{\infty} f(n) ...
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Proof of Meromorphicity of Eisenstein Series in Serre's 'A Course in Arithmetic'

I was reading the chapter on Modular Forms from Serre's 'A Course in Arithmetic'. In the first line of his proof that the Eisenstein Series $G_k(z)$ is a modular form, he says, "The above arguments ...
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Meromorphic functions fixing $\mathbb S^1$

The following question is similar to this one: Determine all meromorphic functions $f \colon \Bbb C \to \Bbb C$ that $$ \vert f(z) \vert = 1 \qquad \forall z \colon \vert z \vert = 1. $$ ...
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Meromorphic functions on upper half plane and infinity

Is the set of all meromorphic functions on the upper half plane which they are also meromorphic at infinity, a field? I am trying to show that the set of all modular functions for $Sl_2(\mathbb{Z})$ ...
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Statement about entire functions

In Stein and Shakarchi's Complex Analysis, there is the following statement: if two entire functions, say $f_1 $,$f_2$, vanish at all $z=a_n $ and nowhere else, then $\frac {f_1}{f_2} $ has removable ...
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Holomorphy of a series of complex functions

I have to discuss the holomorphy of the function $$f (z)=\sum_{n=1}^\infty \frac {e^{nz}} {z-n}, $$ and study its singular points. I would say that in the part of the complex with $\mathrm {Re} (z) \...
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Cohomology of the sheaf $\mathcal{M}^*$ of invertible meromorphic functions on a Riemann surface

Let $X$ be a Riemann surface and denote by $\mathcal{M}^*$ the sheaf of invertible (i.e. not constantly zero on any connected component) meromorphic functions. I have seen claims that $H^1(X, \...
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Convergence of a series of holomorphic functions

A frequent type of exercise, in my complex analysis course, consists of determine where a series of holomorphic functions converges to a holomorphic function. For example: $$f (z)=\sum_{n=0}^\infty\...
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Showing that a given meromorphic function is a rational function and finding its exact form

Consider $f$ holomorphic on $\mathbb{C}\backslash\{1,i\}$. In $z=1$ the function $f$ has a pole of order 5 and in $z=i$ a pole of order 6. Moreover, $f(z)=g(z)^{11}$ holds on $B(0,1/2)$ for some ...
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Prove that if a meromorphic function is constant in an open region then it is constant in all its domain

I am trying to prove that if a meromorphic function is constant on an open subset $U$ then it is constant in all its domain $\Omega$. My idea is: Let $F$ be meromorphic in $\Omega$. Then we have $F=\...
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Suppose that for $\theta \in R$ we have $f(z)=g(z,\theta)$. Now fix $z=z_0$, is $g(z_0,\theta)=f(z_0)$ for all $\theta$ in its domain?

Let $f(z)$ and $g(z,\theta)$ be complex functions with $f$ analytic and $g$ meromorphic. Suppose that for $\theta \in R$ we have $f(z)=g(z,\theta)$. Now fix $z=z_0$, is $g(z_0,\theta)=f(z_0)$ for ...
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How can $\log\zeta(s)$ be meromorphically continued to $\Re(s)>0$?

I'm trying to understand how the function $P:\{s\in\mathbb{C}\ |\ \Re(s)>1\}\to\mathbb C,\ s\mapsto\sum_{p\text{ prime}}\frac{1}{p^s}$ can be continued meromorphically to the half-plane $\{s\in\...
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Multiplication of meromorphic functions

Studying from Conway's book for a qualifying exam, I found an exercise that ask you to prove that if $G$ is a region, the set of meromorphic functions in $G$ is a field. When I tried to prove that $...