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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Special definition of “algebraic curve” for Riemann surfaces?

On the book Algebraic curves and Riemann Surface by Rick Miranda, page 169, I see the following definition: (Last part of definition 1.1) A complex Riemann surface $X$ is an algebraic curve if the ...
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Why are certain contour integrals of doubly periodic meromorphic functions zero?

Consider the following theorem: suppose $f:\mathbb C \to \mathbb C_{\infty}$ is a non-constant doubly periodic meromorphic function. (So, $f(z) = f(z+ \omega_1) = f(z+\omega_2)$ for all $z \in \mathbb ...
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How do I see that this function has removable singularities?

I have the following problem: Let $f:\Omega \rightarrow \Bbb{C}$ be a meromorphic function in a region $\Omega$ with finately many zeros $z_j$ of order $m_j$ and finately many poles $p_k$ of order $...
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When does a Fuch's type 2nd order ODE not have a singularity at infinity?

I know that any second order linear ode $$w''+p(z)w'+q(z)w=0$$ is of Fuchs type (ie, coefficients meromorphic with singular points at $z_0,...,z_n,\infty$ all regular) if $$p(z)=\sum\frac{p_j}{z-z_j}, ...
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Meromorphic function which is analytic bounded on $\mathbb H$, has no zeroes on $\mathbb H^-$, modulus 1 on $\mathbb R$

I am interested in the class $G$ of meromorphic function $f:\mathbb C \to \mathbb C$ such that: $f$ is analytic and bounded on the upper half plane $\mathbb H$. $f$ has no zeroes on the lower half ...
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Meromorphic function with poles of unbounded order

Let: $$ f(z) = \sum_{j=1}^{\infty} \frac{1}{(z - j)^{j}} $$ so that $f$ has a pole of order $j$ at $z = j$ for each integer $j > 0$. Is $f$ meromorphic in the extended complex plane? It is analytic ...
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The quotient of homogeneous polynomial on a smooth projective curve is meromorphic if the denominator is not identically zero.

Suppose $X \subset \mathbb{CP}^n$ is a smooth projective curve. Why is $G/H$ a meromorphic function on $X$ if $G$ and $H$ are homogeneous polynomials of degree $d$ and $H$ does not vanish identically ...
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To check sum of two function is meromorphic or not

$\phi(z) = Re(z) + f(z) $ where $f(z)$ is a meromorphic function. then WOTF is true, 1)$\phi $ is meromorphic 2)$\phi $ and $ f(z) $ has same number of singularities 3)$\phi $ is analytics in every ...
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Meromorphic function inequality

I'm studying "Meromorphic Functions" by W. K. Hayman and I'm stuck with the proof for theorem 1.7.1 (p.18): If $f$ is regular for $|z|\leqslant R$ and $$M(r,f)=\max_{|z|=r}|f(z)|,$$ then $$...
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Meromorphic functions in projective space

It is well known that meromorphic functions in $\mathbb P^n_\mathbb C$ are of the form $p/q$ where $p$, $q$ are homogeneous polynomials in $\mathbb C[x_0 , \ldots , x_n]$ of the same degree, and $q \...
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Embedding of complex torus into $\mathbb{P}^3$

I'm dealing with Riemann Surfaces and I saw how a basis for the space $L(D)$ (where $D\in Div(X)$ is a divisor for the complex torus $T=\mathbb{C}/\Lambda$) of D-bounded meromorphic functions give ...
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Power series with radius of convergence two.

Let $f(z)$ be a power-series (with complex coefficients) centered at $0 \in \mathbb{C}$ and with a radius of convergence 2 . Suppose that $f(0)=0$. Choose the correct statement(s) from below:\ $f^{-1}...
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Holomorphic function that has a triple pole at $0$, a simple pole at $1$, an essential singularity at $i$ and at $-i$

Find a holomorphic function that has a triple pole at $0$, a simple pole at $1$, an essential singularity at $i$ and at $-i$. I know that $f(z)=\frac{1}{z^3}+\frac1{z-1} + e^{\frac{1}{z^2+1}}$ is such ...
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Meromorphic function with non-negative real-part is in fact analytic/holomorphic

Let $f$ be a meromorphic function on an open set $\Omega$ satisfying $Re~f \ge 0$. Prove that $f$ is in fact holomorphic. I'm not entirely sure how to solve this, but here are some thoughts. Let $z_0$...
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Possible values of $l(np)$ for $p\in X$, where $X$ is a compact Riemann surface of genus 2

Here $l(D)$ denotes the dimension of the complex vector space $$\mathcal{L}(D)=\{f\colon X \to \mathbb{P}^1\mathbb{C} \hspace{5pt} \mathrm{meromorphic} \hspace{5pt} \mathrm{s.t.} \hspace{5pt} (f)+D \...
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Why is open mapping a geometric property?

I was recently reading Chapter 3 on meromorphic functions in "Complex Analysis by Stein and Shakarchi". They mentioned that the property that holomorphic functions are open mappings is a ...
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Are meromorphic functions generally (anti)conformal

All analytic functions are conformal whenever their derivatives aren’t $0$. $x^{-1}$ is anticonformal (preserves angles but changes orientations) whenever its derivative isn’t $0$ or $\infty$ Are ...
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Mittag-Leffler Theorem Correction Factor

From what I learned and also in http://math.iisc.ac.in/~vvdatar/courses/2020_Jan/Lecture_Notes/Lecture-17.pdf, it is said if $\Omega=\mathbb{C}$, then $q_k$ can be polynomial. I wonder how if $\Omega \...
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Meromorphic continuation of L-functions

I am following these notes and on page 2 the claim is that if we have an $L$-function $$L(s) = \sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ with $a_n=O(n^r)$ and if $L$ has a meromorphic continuation and ...
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Convergence of sequence of meromorphic functions with moving poles

I'm interested in the notion of convergence of a sequence of meromorphic functions by moving the poles. I'm taking a first course in (graduate) complex analysis, coming from a real analysis background....
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Is $f(z)=z$ a meromorphic function?

Consider a function $f:\mathbb{P}^1\to\mathbb{C}$ s.t. $f|_U(z)=z$ on $U=\{[x:y]\in \mathbb{P}^1|x\neq0\}$ where $z=\frac{y}{x}$ and $f|_V(w)=w$ on $V=\{[x:y]\in \mathbb{P}^1|y\neq0\}$ where $w=\frac{...
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How can I find a closed form for the infinite product $f(z) = z\prod_{k=1}^{\infty} (1-z^3/k^3)$?

Clearly, we have that $f(z) = z + \sum_{k=1}^{\infty} a_k z^{3k+1}$. Incidentally, it turns out that the function $g(z) = \sin(\pi z)e^{\pi z/\sqrt{3}}$ has exactly the same zeros as $f(z)$ [note ...
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Does great Picard's theorem for meromorphic function implies that for entire function.

I have studied the following version of the Great Picard theorem: Suppose $f(z)$ is meromorphic on a punctured neighborhood $\left\{0<\left|z-z_{0}\right|<\delta\right\}$ of $z_{0}$. If $f(z)$ ...
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3 votes
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Gamma function as the ratio of two holomorphic functions

Prelude Be $U \subset X$, where $U$ is open and $X$ is a complex manifold. A meromorphic function $f$ over $U$ is a holomorphic function over $U \backslash S$ (Where $\bar{S}$ has no interior) such ...
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Understanding complex differentiability at infinity

I'm looking for resources that explain the definition of complex differentiability at infinity via the complex projective line. Could anyone suggests such a resource? Many thanks!
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How to improve the precision of a meromorphic approximation iteratively?

Background : The other day I managed to find a numerical approach to estimate the position of poles and zeroes of a Meromorphic function. The issue is that sometimes it does not converge perfectly it ...
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Is there a connection between the analytical properties of complex analytic functions and the complex numbers being complete algebraically?

I will first have to apologize that this will be a very fuzzy question. At this time I have no better way to formulate it. I am willing to reformulate it when I can better pinpoint what it is that ...
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Doubt in Residue Theorem and Meromorphic function on Sphere

I have come across the following two statements. By Riemann Roch theorem on $\mathbb CP^1$ due to g=0, we have dim $H^0(P)=2$ where P is seen as the divisor $1.P$, but $H^0(0)=1$, here 0 is the ...
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2 votes
1 answer
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If $f$ is meromorphic in some region, why can't $\frac{f'}{f}$ have any essential singularities in that region?

It is stated in the Wikipedia article on the argument principle that the only singularities of $\frac{f'}{f}$ are the zeros and poles of $f$ itself. It is not at all clear that there are no essential ...
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Existence of a meromorphic extension of special sums of monomials

Let $(f_k)_{k \in \mathbb{N}_0}$ be a sequence in $\mathbb{R}$ s.t. $\lim_{k \to \infty}f_k =0$. Consider $$ F(z):=\sum_{k\geq 0} z^{k}f_k.$$ Of course $F$ is holomorphic on the unit disk. If $f_n =1\...
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Divisor of meromorphic functions on compact Riemann surfaces

Lemma. If $f$ is a nonzero meromorphic function on a compact Riemann surface, then $\deg\left ( \operatorname{div}\left ( f \right ) \right )=0.$ Proven in Rick Miranda's book, page 130. Question. Is ...
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Decomposition of a meromorphic function on the torus into the Weierstrass p-function and two automorphisms

Say $\tau $ be a complex number with $\Im(\tau)>0$.Then a meromorphic function $f$ on the torus $M$ can be thought of as a meromorphic function on the plane with $f(z+1)=f(z)$ and $f(z+\tau)=f(z)$ $...
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Meromorphic section of a subbundle

I have recently been trying to extend my knowledge of complex analysis by expanding to Riemann surface theory. I had been recommended Forster's Lectures on Riemann Surfaces as a good reference to go ...
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Do Meromorphic Functions automatically give us Entire Functions?

Suppose I have a meromorphic function, $f$. Then, I can write $f(z)=\frac{h(z)}{g(z)}$ where $h,g$ are entire. I would really like to be able to claim that $f$ extends to an entire function by ...
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Residues of $df/(f-a)$ for meromorphic $f$.

Suppose $f$ is a non-constant meromorphic function on a compact Riemann manifold and $a\in \mathbb{C}$. I'm trying to understand why $$\mathrm{res}_P\left(\frac{df}{f-a}\right) = \frac{1}{2\pi i }\...
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The closure of meromorphic functions under composition

It is well-known that composing meromorphic functions on $\mathbb C$ does not necessarily result in a meromorphic function (e.g., $\exp\circ\frac1x$, which has an essential singularity at $x=0$.) ...
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Why does Mittag-Leffler's theorem work over the entire complex plane?

I have a minor factual clarification about the statement of Mittag-Leffler's theorem, and also a more substantive question about its proof. Q1. The Wikipedia article on Mittag-Leffler's theorem states ...
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$H(G) \cup \{\infty\}$ is closed in $C(G,\mathbb C_{\infty}).$

Let $G \subseteq \mathbb C$ be a region and $H(G),M(G)$ and $C(G,\mathbb C_{\infty})$ be respectively denote the spaces of holomorphic functions, meromorphic functions and extended complex valued ...
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2 votes
1 answer
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Prove $g(z)=\frac{f'(z)}{f(z)}$ where $g$ has simple poles with entire residue.

Let $g$ be a meromorphic function on $\mathbb{C}$. Suppose $g$ has poles of order 1 with integer residue. Prove that exists $f$ meromophic such that $$g(z)=\frac{f'(z)}{f(z)}.$$ I was able to prove it ...
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Extending analytically $\sum_{n=1}^\infty \frac{1}{e^{1/z}+e^n}$ in $0$.

The function $$S(z)=\sum_{n=1}^\infty \frac{1}{e^{1/z}+e^n}$$ is meromorphic in $\mathbb{C}\backslash \{z_{n,m}=\frac{1}{i\pi(2m+1) +n}:n\in \mathbb{N},m\in \mathbb{Z}\}$ with simple poles at every $...
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Meromorphic function with $n+\sqrt{n}$ as poles of second order.

Find a meromorphic function with poles at $z_n=n+\sqrt{n}$ with $n\in\{1,2,3,\ldots\}$ such that $$f(z)-\frac{1}{(z-z_n)^2}+\frac{1}{z-z_n}$$ has a removable singularity at $z_n$. I believe that I ...
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Proving that a mermorphic function with three linearly independent (over the rationals) periods is constant

I would like to understand and answer the following question from Serge Lange's Introduction to Complex Analysis at a graduate level. I understand how is one supposed to use the hint to prove the ...
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Meromorphic 1-form on $\Bbb C_\infty$ with prescribed poles and residues

Let $X=\Bbb C_\infty$ be the Riemann sphere $C\cup \{\infty\}$. Given finitely many points $p_1,\dots,p_n\in X$ and a corresponding complex numbers $r_1,\dots,r_n$, can we construct a meromorphic $1$-...
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Descent of holomorphism also holomorphic - hyperelliptic Riemann surfaces

My question is about how to finish a proof of Lemma 1.9 from Rick Miranda's Algebraic Curves and Riemann Surfaces. Essentially, a hyperelliptic Riemann surface is the solution set of the equation $y^2=...
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Substitution Rule for Contour Integral of Meromorphic Function

Given a meromorphic function on a domain $D\subset\mathbb{C}$. Is there a notion of substitution rule for a contour integral over the boundary $\partial{D}$? And what are the needed properties of the ...
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Meromorphic function on punctured unit disk with poles at $\frac{1}{n}$ is dense in $\mathbb C$ arbitrarily close to zero

Let $f$ be meromorphic in the punctured unit disk with poles at the points $1/n$, $n =1, 2, 3, \ldots$. I want to show that the image of an arbitrarily small neighborhood of zero under $f$ is dense in ...
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Why are Cousin problems so relevant?

It is known that Cousin problems are especially relevant in determining if a meromorphic function can be expressed as a quotient of two holomorphic functions. Indeed: Theorem 1 Let $X$ a complex, ...
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Holomorphic function such that $\theta\left(z+\omega_j\right)=a_j\theta\left(z\right)$ satisfies $\theta\left(z\right)=ae^{bz}$.

I am trying to solve the following exercise: Let $\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ be a lattice in $\mathbb{C}$ and let $\theta$ be an entire function such that there exist $a_1,a_2\in\mathbb{C}...
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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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3 votes
1 answer
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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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