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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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f is a meromorphic function satisfying $\vert f(z)\vert\leq\vert z\vert^n$ then f is a rational function

I would like to see how the following question can be proved: Let $f\in\mathcal{M}(\mathbb{C})$ satisfying $\vert f(z)\vert\leq M\vert z\vert^n$ for all $z\in\mathbb{C}\setminus P(f)$ with $\vert z\...
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Meromorphic function $f$ with finite set of poles: existence of rational function $h$ s.t $p(h)=p(f)$?

Show that if $f$ is meromorphic in $D$ and has a finite set of poles, then there is a rational function $h$ with $p(h)=p(f)$ and $(f-h){\vert_D}\in\mathcal{O}(D)$ This is taken from Remmert's Theory ...
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Meromorphic Function on a Riemann Surface

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at pages 73/74): In the proof we construct locally in $U$ a polynomial $...
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Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
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Zeros of meromorphic function on Riemann surface and paracompactness

I am looking at an exercise at the end of this video on YouTube about Riemann surfaces. We are asked to show that for a nonzero complex-valued meromorphic function $f$ defined on a connected open ...
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Laurent series and tensor

Let us begin with the complex vector space \begin{equation} V_{z}=\Big\{\omega\in \mathbb{C}[[z,z^{-1}]]dz\ \mid \operatorname{Res}_{z=0} \omega (z)\Big\} \end{equation} We could define the tensor ...
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When is composition of meromorphic functions meromorphic

When I compose a meromorphic and a holomorphic function, I get a meromorphic function. Are there other cases when a composition of two meromorphic functions is meromorphic? For example, if I compose a ...
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$\frac{z}{e^z-1}$ partial fractions

How do I expand $$\frac{z}{\mathrm{e}^z-1}$$ into partial fractions using Mittag-Leffler theorem? If I already know the decoposition of $$\frac{1}{\mathrm{e}^z-1}$$ how, if at all, can I use it to ...
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How to prove that $\frac{1}{1-z}=\prod_{n=0}^\infty (1+z^{2^n})$?

A simple pole can be written as $\displaystyle{\frac{c}{c-z}=\prod_{n=1}^\infty e^{\frac{1}{n}\left(\frac{z}{c}\right)^n}}$. How does one show that when $c=1$, $\displaystyle{\frac{1}{1-z}=\prod_{n=0}^...
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Weak solutions for divisors

I have a question on the following definition in the Forster: I don't get the part where it says "Clearly a weak wolution $f$ is a proper, i.e., meromorphic function, solution precisely if $f$ is ...
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Example of a complex meromorphic functions which is not locally Lipschitz [closed]

Let $U\subset \mathbb{C}^n$ be an open subset and let $\varphi$ be a meromorphic function on $U$ i.e. locally given by the ratio of two holomorphic functions. Can $\varphi$ be not locally Lipschitz?
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Cousin I problem in $\mathbb{C}$ and Mittag-Leffler theorem

In the wikipedia page about Cousin's problems (https://en.wikipedia.org/wiki/Cousin_problems) is stated that "the case of one variable is the Mittag-Leffler theorem"; I'm a bit in trouble with this ...
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Where does this “factorization” of a meromorphic function come from?

I encounter a remark in reading a book which states: If $f(x) \in \mathbb R[x]$ is a monic polynomial, i.e., $f(x) = x^n + a_{n-1} x^{n-1} + \dots + a_0$, then \begin{align*} \frac{ f'(x) }{f(x) }...
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Is the domain of a complex function always open? Is $\mathbb C$\ (the domain) always of measure zero? What if the function is holomorphic?

Usually the domain of a complex function is $\mathbb C\backslash\{z\in\mathbb C~:~z \text{ is a singularity of } f \}$ So I guess it must always be $\mathbb C\backslash\{\text{a set of points}\}$. ...
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Showing algebraic dependence of meromorphic functions on a compact Riemann surface

I have been given the following question to do: Let $f,g$ be meromorphic functions on a compact Riemann Surface $R$. Show that there is some polynomial such that $P(f,g) = 0$ (i.e. show that any two ...
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Why is the derevative of meromorphic function is meromorphic?

I know that if $f$ is meromorphic then $\exists A\subset \Omega$ s.t $f$ is holomorphic on $\Omega \setminus A$ and $A$ is discrete, and $A$ are the poles of $f$. I want to show that $f'$ is ...
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Proof that for every Meromorphic function, it`s derevative is also Meromorphic

for $f(z)$ Meromorphic we know that $f(z)$ on $\Omega$ - an open set ,is Holomorphic on $\Omega/A$ when $A$ is discrete set of the poles of $f(z)$. I know that $f'(z)$ is also Holomorphic on $\Omega/...
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Given a meromorphic function, $D_{r+\delta}(w)\backslash D_r(w)$ has no poles or zeros.

Given a meromorphic function $f$ on $\Omega$ that is not identically zero, and $f$ has no poles and never vanishes on $\partial \overline{D_r(w)}\subseteq \Omega$. I'm trying to prove that there ...
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Analytic continuation of quotient of analytic functions

Suppose $f(z)$ and $g(z)$ are defined for some open subset $U$ of the complex plane, and that they are holomorphic on that subset. We then know that their pointwise quotient $(f/g)(z)$ is meromorphic ...
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Is there a function with prescribed zeros and poles on an elliptic curve?

Let $T$ be the complex tore from the lattice $(1, \tau)$ where $im(\tau)>0$. How to prove the existence of a meromorphic function on $T$, with divisor $(0) + (\frac{1}{2}) - 2 (\frac{\tau}{2})$ ? (...
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Why are these called “meromorphic” differential forms?

On p.30 of Silverman's The Arithmetic of Elliptic curves, the notion of a meromorphic differential form on a curve $C$ is defined as follows: Definition. Let $C$ be a curve. The space of (...
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Is $\frac{w-1}{w+1}$ a conformal mapping from $\mathbb{C}\setminus(-\infty,0]$ onto $\mathbb{C}\setminus\big((-\infty,-1]\cup[1,+\infty)\big)$?

Is the meromorphic function $g$ defined by $g(w) := \frac{w-1}{w+1}$ a conformal mapping from the singly slit plane $\mathbb{C}\setminus(-\infty,0]$ onto the doubly slit plane $\mathbb{C}\setminus\big(...
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About the proof of $\sum\limits_{n=-\infty}^\infty f(n)=-\pi\sum\limits_{k=1}^m\text{res} [f(z)\cot(\pi z)]_{z=a_k}$?

Let $f(z)$ be a meromorphic function with a finite number of poles $a_1,\dots,a_m$, where $a_i\not\in\mathbb Z\cup\{0\}$. Prove that if there exists a sequence of contours $\{C_n\}$ that goes to ...
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Polynomial equation with meromorphic coefficients and implicit function theorem

In his article Integration in Finite Terms, Maxwell Rosenlicht writes that "the implicit function theorem shows that if we are given a polynomial equation with coefficients which are functions ...
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Showing a holomorphic function on $\mathbb C-{0}$ satisfying $|z\frac{du}{dz}|\le c|u(z)| $ is meromorphic

I am looking for a proof of the following argument, which comes from a book: Proposition: Let $u$ be a holomorphic function on $\mathbb C-{0}$, and assuming that there exists $c>0$ such that for ...
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Why are nonzero eigenvalues of a compact operator poles of its resolvent?

Let $X$ be a complex Banach space and $T$ a compact operator on $X$. I read on Wikipedia that nonzero elements of the spectrum of $T$ are poles of its resolvent. It says "by functional calculus", but ...
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Terminology for a set of meromorphic functions with controlled poles and orders

I wonder if there is a name for a family $F$ of meromorphic functions $f : \mathbb C \to \mathbb C$ all of whose poles are contained in a fixed closed discrete set $S = (s_i)_{i \in \mathbb N}$ with ...
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Residue of product of Meromorphic function with $e^{\lambda}$

If $f(\lambda)$ is meromorphic then is it true that it holds for the residue that $\operatorname{Res}_{\lambda=\lambda_0}(f(\lambda)e^\lambda)=\operatorname{Res}_{\lambda=\lambda_0}(f(\lambda))e^{\...
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Why does a meromorphic function with limit at infinity continue to have limit at infinity when the poles are removed?

I wanna prove the following: If I have a meromorphic function $f$ with finitely many poles $p_{1 }, ..., p_{n}$, and it has a limit at infinity (potentially being infinity, as in, the function is ...
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What is known about the reciprocal $1/f$ of a holomorphic Banach-valued function $f$?

Let $U \subseteq \mathbb C$ be open, $A$ a (unital, associative) complex Banach algebra and $f : U \to A$ holomorphic and invertible in a punctured neighborhood of $0 \in U$, so that $0$ is an ...
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Calculate $\sum_{n=1}^\infty\frac{1}{n^2}$ and $\sum_{n=1}^\infty\frac{1}{n^4}$

I need to do that using $$\sum_{n \in \mathbb{Z}}\frac{1}{(z-n)^2}=\left(\frac{\pi}{\sin \pi z}\right)^2$$ I've already prove that this is true. The thing is that this function in meromorphic and ...
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Meromorphic function with poles along a hypersurface

Let $X$ be a complex variety and $D \subset X$ an hyperusrface. We say that a function $$f: X-D \to \mathbb{C}$$ is meromorphic along $D$ if for every $p \in D$ there exixsts $V_p \ni p $ open subset ...
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What are the possible poles of the meromorphic function $G$?

Consider the function $$F(z)=\int_{1}^{2} \frac {1} {(x-z)^2}dx,\ \mathrm {Im} (z) > 0.$$ Then there is a meromorphic function $G(z)$ on $\Bbb C$ which agrees with $F(z)$ when $\mathrm {Im} (z) >...
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How to construct the function on torus with two distinct simple poles?

Consider torus $T=C/\Lambda$ where $\Lambda$ is a rank 2 lattice and $C$ is complex plane. Then $g(T)=1$. From riemann roch, I deduce that $P\neq Q, h^0(P+Q)=2+(1-1)+0=2$ where $0=h^0(K_T-P-Q)$ by $...
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a question about specific entire function

Find all the entire functions $f(z)$ for which there exists positive constants $M, \:R$ and a positive integer $n$ such that $$|f(z)|\geq M.|z|^n\:\:\text{whenever}\:\:|z|>R.$$ What if the ...
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Showing $f(z)$ is a rational function

I could prove that if $g(z)$ is bounded and analytic except at a finite number of points in the complex plane $\mathbb{C}$, $g(z)$ must be constant. First I thought I can use the above to prove the ...
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Given an inequality of two meromorphic functions, prove that they are equal up to a constant.

The following is a past qualifying exam question in complex analysis: Let $f$ and $g$ be two meromorphic functions in $\mathbb{C}$. Assume that $|f(z) + g(z)| \le |g(z)|$ for all $z \in \mathbb{C}$...
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Meromorphic functions on $\hat{\mathbb{C}}$ are rational functions

I was reading through a proof of the fact that all meromorphic functions on $\hat {\mathbb{C}}$ are rational functions found here http://math.haifa.ac.il/hinich/RSlec/lec1.pdf and I didn't ...
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Extending $\zeta$ to a meromorphic function on $\mathbb{C} - \{1\}$

I know that we can extend $\zeta (s)$, originally defined on $\Re(s)>1$ by the sum $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$, to the domain $\mathbb{C} - \mathbb{Z}$ by this definition: $$\zeta(s)...
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Upper bound for residue of a rational function with zeros and poles in the disk

Let $r(z)$ be a monic rational function (that is, a ratio of monic polynomials) all of whose finite zeros and poles lie in the unit disk of degree $n$. QUESTION: Is there a nice upper bound on the ...
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Reciprocal Zeta function: between 0 and 1/2? [closed]

According to resources from the internet, if Riemann Hypothesis is true, then $$\frac1{\zeta (z)}=\sum^\infty_{n=1}\frac{\mu (n)}{n^z}$$ converges for $1/2<\sigma$. However, I cannot find any ...
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Integral on semicircular arc of a function with a simple pole at $z=0$

Let $C(\epsilon)$ denote the semicircular arc about $z=0$. Suppose $f(z)$ is a meromorphic function that has a simple pole at $z=0$. I'm trying to show that $$\lim_{\epsilon\to 0}\int_{C(\epsilon)} f(...
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Mittag-Leffler Theorem Exercise - Find Sum of Series [duplicate]

The exercise asks to find the closed form of $$\sum_{n=-\infty}^{\infty}\frac{1}{(z+n)^2+a^2}$$where $a$ is a complex number. I know that $\frac{\pi^2}{\sin^2(\pi z)}=\sum_{n=-\infty}^{\infty}\frac{1}...
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Sheaf of Meromorphic Functions

In one of the exercise sheets for my complex analysis course we are given the following task Prove that the set of all meromorphic functions on $\mathbb{C}$ defines a (pre)sheaf. showing that they ...
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representation for a pole of order N

If a meromorphic function has a single pole of order $N$ at $z_0$, then is the principal part of that function always $$\frac{c}{(z-z_0)^N}$$ for some constant $c$? A counterexample or proof would be ...
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Meromorphic Functions that satisfy a first order algebraic differential equation

I have been writing a project on elliptic functions and in this project I prove the following theorem: Suppose a meromorphic function $f$ satisfies an algebraic addition theorem, that is there exists ...
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Fourier transform of meromorphic function

Suppose that I have a function $f(z)$ which is meromorphic on the entire complex plane, meaning holomorphic everywhere except for a discrete set of poles. I then take a vertical slice of this ...
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Existence of meromorphic root for meromorphic function

The exercises states: Let $f:D \to \mathbb{C}$ be a meromorphic function with a finite number of roots and poles, $D$ being some simply-connected open set. Prove that there exists a meromorphic ...
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Can a meromorphic function have removable singularities? [duplicate]

Authors usually give the following definition (or some variation) of meromorphic function: Let $G$ be a region (i.e. open connected subset of $\mathbb{C}$). A function $f$ is said to be ...
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Image of a neighborhood of a pole

Let $f$ be a meromorphic function and $a$ a pole of $f$. Let $U$ be a neighborhood of $a$. Is it true that $f(U)$ contains $\{z| |z|>r\}$ for some $r$? All I know is that $\lim\limits_{z\...