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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Why is the order of $dz$ on Riemann sphere at $\infty$ equal to $-2$? [duplicate]

The setting is as follows: We consider $X= \mathbb{C}_\infty$, the Riemann sphere with coordinate $z$ on $\mathbb{C}$. Let $\omega = dz$ (this is a $1$-form). On page 131 of Rick Miranda's book "...
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Applying the Identity Theorem to Analytic Functions Agreeing on 1D Curves

I'm working through a complex analysis problem and have encountered a problem that I'm struggling to understand. The context involves two meromorphic functions, g and h, which are given on the unit ...
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Characterizing functions that satisfy the reflection $f(z)f(-z)=1$

Context: During a mathematical discussion with a good friend, I was brought to think about the functional equation $q(t)q(1-t) = 1$ for $q$ with some regularity. (To be honest the exact context doesn'...
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Why is the quotient of two local defining functions for divisors nowhere vanishing?

I am trying to follow Griffiths & Harris, "Principles of Algebraic Geometry", page 131, where they explain how from a given divisor one obtains an element of the quotient sheaf $\mathcal ...
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Understanding the claim that holomorphic divisors contained in a divisor class are in bijection with the projectivisation of the space of sections

Let $V$ be a complex manifold. I am trying to approach the subject of divisors through complex geometry since I am not familiar enough with algebraic geometry to come that way. A divisor $D$ on $V$ is ...
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Meromorphic function with infinitely many poles must have essential singularity at point of infinity?

Question: Suppose $f: \mathbb{C}\setminus P \to \mathbb{C}$ is meromorphic on the complex plane, where $P:=\{z_n: n\in\mathbb{N}\}$ is a set of infinitely many isolated poles with $\infty$ as their ...
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On the topic of Exercise 14.4 question 9 in Analytic Function Theory Vol. II by Einar Hille

So I have the book Analytic Function Theory Vol. II by Einar Hille and have been working through the exercises as I go. Now, let's take a look at Exercise 14.4 question 8 (since this ties into ...
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Holomorphic function on unit disc with specific properties [duplicate]

I am supposed to characterize all holomorphic functions on the unit disc on to the complex plane, such that there exists $N\in\mathbb{N}$ such that for all $n>N$: $\lvert f(\frac{1}{n})\rvert\leq e^...
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Meromorphic infinite product mapping upper half-plane to itself [duplicate]

Given 2 sequences $(a_n)_{n\in\mathbb{Z}}$, $(b_n)_{n\in\mathbb{Z}}$ with $b_n<a_n<a_{n+1}$, and $a_{-1}<0<b_1$ I'm trying to show that the function defined by $$\theta(z)=\frac{a_0-z}{b_0-...
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Transcendence of meromorphic function vs formal power series

Consider the meromorphic function $f$ on $\mathscr D=\{z\in\mathbb C\mid|z|<1\}$ definied by $\displaystyle f(z)=\sum_{n\ge1}\frac{z^{2^n}}{z^{2^n}-\frac12}$. Obviously $f$ admits infinitely many ...
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Complex function with specific residues

I am supposed to find a holomorphic function $f:\mathbb{C}\setminus \{-1,+1\} \to \mathbb{C}$ such that it has essential singularities in $-1$ and $+1$ and $res_{-1}f=-1$ and $res_{+1}f=+1$. Is this ...
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Limit of X(s)X*(s)/(X(s)-X*(s)) as X(s) approaches zero for Meromorphic X(s)

Let $X(s): X,s \in \mathbb{C}$ be meromorphic, with simple, isolated zeros $\{z_n\}$ and first-order, isolated poles $\{p_n\}$, and let $L_k=\lim_{s\to z_k} \frac{X(s)X^*(s)}{X(s)-X^*(s)}$. Is $L_k=0$?...
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Quotient of meromorphic functions

Can anyone please explain me this highlighted line- My intital thoughts are as follows. If $f$ and $g$ have a common zero say $z_0$ then, if $$lim_{z\to z_0}(z-z_0)\frac{f(z)}{g(z)}=lim_{z\to z_0}(z-...
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An automorphic function with no poles is constant.

Daniel Bump calls $f$ an automorphic function if it satisfies the formula $$f\left(\frac{az+b}{cz+d}\right)=f(z)$$ where $\begin{pmatrix} a&b\\ c&d \end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$ ...
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Every holomorphic map into Riemann sphere is meromorphic

Let $\mathbb C^* := \mathbb C \cup \{\infty\}$ be the Riemann sphere. I know that every meromorphic function $f \colon \Omega \to \mathbb C^*$ with $\Omega \subseteq \mathbb C$ open can be seen as a ...
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Does the order of vanishing at infinity for an automorphic form depend on the chosen period?

In Diamond and Shurman's A First Course in Modular Forms, the authors define meromorphy at infinity as follows: Let $f$ be a meromorphic function on the upper half plane that is weakly modular with ...
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Showing that a meromorphic function has a dense range in a punctured disk with a non-isolated singularity

Suppose that $f$ is a meromorphic function on the punctured disk $0<|z|<1$ having poles on the set $\left\{\frac{1}{k}: k\in\mathbb{N}\right\}$. I would like to show that for every $1>r>0$,...
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Looking for a specific zeta function.

I am looking for a zeta function $$ f(s) = \sum \frac{1}{a_n^s}$$ Where $a_n$ is a sequence of distinct positive integers, such that $f(s)$ is analytic for all $Re(s) > 1$ $f(s)$ has a simple ...
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Advantage of an alternative formula for complex gamma function

We know that for $Re(z) > 0$ we have per definition$$\Gamma(z) = \int_{0}^{\infty}e^{-t}t^{z-1}dt$$ and this is well defined for the domain. If we extend the function into a unique meromorph ...
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Suppose $f(z)$ is meromorphic on $\mathbb C$, and there are two circles $K, K'$, so that $f(K)⊂K'$, prove that $f(z)$ is a rational function.

Suppose $f(z)$ is meromorphic on $\mathbb C$, and there are two circles $K$, $K'$, so that $f(K)⊂K'$, prove that $f(z)$ is a rational function. I think $f(z)$ has to extend to $\mathbb{C}∪\{∞\}$ and ...
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Extending a holomorphic function $\mathbb{C} \rightarrow \mathbb{C}$ to be holomorphic on the Riemann sphere $\mathbb{C}^{*}$

How can I quickly see if a function $f:\mathbb{C} \rightarrow \mathbb{C}$ can be extended to $f:\mathbb{C}^* \rightarrow \mathbb{C}^*$? (where $\mathbb{C}^*$ denotes the Riemann surface $\mathbb{C} \...
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Prove that the any Riemann surface of genus zero is isomorphic to the Riemann sphere

Let $X$ be a compact Riemann surface, and suppose that there exists a meromorphic function $f$ on $X$ with one simple pole and no other poles. Show that $f$ is an isomorphism between $X$ and the ...
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Find all meromorphic functions $f$ s.t. $|f(z)|=1$ wherever $|z|=1$.

Problem. Find all meromorphic functions $f$ s.t. $|f(z)|=1$ wherever $|z|=1$. I know this problem has been solved in this posting. However, I cannot show $f$ has only finite number of poles and zeros....
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Given two points $P$ and $Q$ on a fundamental parallelogram, construct an elliptic function with simple poles at $P, Q$ by contour integral of $\wp$

We know that every elliptic functions can be written as a rational function of $\wp$ and $\wp'$, where $\wp$ is the Weierstrass p function.. However, I am wondering how, given two arbitrary points $P$ ...
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Question about divisors on a compact Riemann surface

I am trying to prove the following question from here without relying on the Riemann-Roch theorem. Let $X$ be a compact Riemann surface, and let $D$ be a divisor on $X$. (i) If $\mathrm{deg}(D) = 0$, ...
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The degree of a meromorphic form in a genus $g$ Riemann surface is equal to $2g-2$… is this in some way related to the Gauss-Bonnet theorem?

I’m following an introductory course on Riemann surfaces. Today, the lecturer proved the fact that the degree of a meromorphic form in a genus $g$ Riemann surface is equal to $2g-2$ (we can deduce ...
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If two elliptic functions share the same poles and zeros (including multiciplity) then they are proportional

I’m trying to understand the following statement found on my lecture notes: If two elliptic functions share the same poles and zeros (including multiciplity) then they are proportional I’m trying to ...
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The field of meromorphic functions over $\hat{\mathbb{C}}$ is equivalent to the field of rational functions

I’d like to prove the following statement from my lecture notes: The field of meromorphic functions over $\hat{\mathbb{C}}$ is equivalent to the field of rational functions, with the convention $1/0=\...
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A question related to an inequality involving a complex function

I'm trying to solve the following problem from one of the past qualifying exams. Find an explicit formula for all meromorphic functions $g$ on $\mathbb{C}$ such that $$|g(z)|\leq \frac{\log(2+|z|^2)}{...
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Definition of meromorphic $q$-differential

In Farkas Kra riemann surfaces textbook, it defines a meromorphic $q$-differential as follows: Let $q$ be an integer. By a (meromorphic) $q$-differential $\omega$ on $M$ we mean an assignment of a ...
one potato two potato's user avatar
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Meromorphic continuation of $\zeta(s)$ to ${\rm Re}(s)>0$.

Riemann zeta function is one of the most mysterious functions that we encounter in mathematics.We require a meromorphic continuation of this function to ${\rm Re}(s)>0$ in order to prove the prime ...
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What is the definition of sheaf of meromorphic differential form on a complex manifold?

Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is the sheaf of $\mathbb{C}$-valued holomorphic functions on $X$. Let $T_X^\vee$ be the cotangent bundle over $X$ with the ...
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Show that $\frac{f'(z)}{f(z)}$ has a pole if and only if $f(z)$ has a zero or a pole.

In our analytic number theory course, our instructor told us that $\zeta(s)$ has a zero or a pole at $z_0$ if and only if $\frac{\zeta'(s)}{\zeta(s)}$ has a pole at $z_0$. In fact, this is true for ...
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conformal mapping from unit disk to equilateral triangle

Let $$f(z) = C \int_{0}^{z} \frac{1}{(1-\zeta^3)^{\frac{2}{3}}}d\zeta$$ with $$C \int_{0}^{1} \frac{1}{(1-x^3)^{\frac{2}{3}}}dx=1$$ be a conformal map from the unit disc $B(0,1)$ to the equilateral ...
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Power series expansion of 1/(1+z)^2 around z=1 [duplicate]

I'm struggling with getting the power series/Taylor series expansion of $f(z) = \frac{1}{(1+z)^2}$ around $z_0 = 1$. Usually, I would do a partial fraction decomposition, and then do some re-arranging ...
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Solutions of complex linear difference equations

I know solutions of complex linear differential equations like \begin{equation} f^{(k+1)}(z)+c_{k-1} f^{(k)}(z)+\ldots+c_1 f^{\prime}(z)+c_0=0 \end{equation} are linear combinations of $e^{\alpha z}$. ...
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Exercise on Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$ and let $p,q\in X$ be two distinct points. I want to prove that: $$\ell(p+q)\in \{1,2\}$$ Clearly $\ell(p+q)\geq 1$ because it contains ...
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Equality between $\text{gcd}$ and $\text{lcm}$ of divisors on a compact Riemann surface

Notation: If $E$ is a compact divisor on a Riemann surface and $p\in X$, I'll use the notation $E^{(p)}$ to denote the coefficient of the point $p$ in $E$. Let $D\geq 0$ be a divisor on a compact ...
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Difficulties in finding the radius of the disk in which power series of a function is analytic

I am trying to find a function of asymptotic growth of the Fibonacci sequence, which's generating function is $$ f(z)=\frac{1}{1-z-z^2}=\frac{1}{(1-\phi _-z)(1-\phi_+z)} $$ with poles at $ z_1=-\phi_- ...
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Kernel of $\text{Res}_p$ on a Riemann surface

Let $X$ be a Riemann surface and let $p\in X$. Let: $$\text{Res}_p:\Omega^1_{\text{mero}}(X)\to \mathbb{C}$$ be the residue at $p$ operator defined on meromorphic $1$-forms. Clearly holomorphic $1$-...
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Meromorphic function - Countably infinite or finite number of poles?

I am having trouble reconciling the following two questions: Can it be proved that a Meromorphic function only has a countable number of poles? $f$ meromorphic on $\mathbb{\hat{C}}$ $\implies$ $f$ ...
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Schwarz reflection principle for meromorphic functions

I'm trying to work on this exercise from Conway's Complex Analysis textbook (Exercise 3, Chapter 9, Section 1). Here $G$ denotes a symmetric region, $G_+$ the points of $G$ above the real axis and $...
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Why is $\mathcal K_X$ the "sheaf of rational functions"?

Let $f:X\to\mathbb A$ be a regular function on a scheme. Is there some relation between $f$ being a non-zero divisor in $\mathcal O(X)$ and the denseness (topologically or scheme theoretically) of $D(...
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Confusion on a note by Conway on Marty's Theorem

I am stuck on a note by Conway right before the proof of Marty's Theorem (equivalent of Montel's Theorem, for meromorphic functions) in "Function of One Complex Variable I". The family of ...
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How to prove the quotient of homogenous polynomial with same degree is a meromorphic function on $\Bbb{CP}^n$

Let $f,g$ be two homogenous polynomial of degree $d$ in $\Bbb{C}[x_1,...,x_{n+1}]$. define the quotient $f/g$ on $\Bbb{CP}^n$, intuitively it should be the meromorphic function, but I want to prove it ...
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Definition of meromorphic function on complex manifolds

There are typically four(or more?) definition of meromorphic function on Riemann surface and three definitions of meromorphic function on complex manifolds, I want to show they are equivalent (I will ...
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Analytic function and derivatives

Show that a smooth complex function $f(z, \bar{z})$ satisfies $\frac{\partial^2 f}{\partial \bar z^{2}}=0$ if and only if $f(z, \bar{z})=\bar{z}g(z)+h(z)$ for some analytic functions $g(z)$ and $h(z)$....
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Show that $\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$ is meromorphic

Let $p\geq 1$ be an integer. If $\sum_{n=1}^\infty 1/|a_n|^{p+1}$ converges, show that $$\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$$ is ...
one potato two potato's user avatar
3 votes
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Inverse and extension of the $n$th Airy Ai and Bi zero

$\def\ai{\operatorname{ai}} \def\bi{\operatorname{bi}} \def\Ai{\operatorname{Ai}} \def\Bi{\operatorname{Bi}} $ Airy Ai Zero $\ai_n$ gives the $n$th zero of the Airy Ai function and Airy Bi Zero $\bi_n$...
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Can a meromorphic function $f:U \to \mathbb{C}, \; U\subset \mathbb{C}$ domain, that is not the zero function, have a zero of order infinity?

Can a meromorphic function $f:U \to \mathbb{C}, \; U\subset \mathbb{C}$ domain, that is not the zero function, have a zero of order infinity? (Clear: Would $f$ be holomorphic on the whole domain U ...
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