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Questions tagged [meromorphic-functions]

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Behavior of the hypergeometric function near x=1

It is known that the hypergeometric function ${}_2F_1(a, b, c; x)$ defined by the series $$\sum_{n=0}^\infty \frac{a(a+1)\cdots (a+n-1)\cdot b(b+1)\cdots (b+n-1)}{c(c+1)\cdots (c+n-1) n!}x^n$$ behaves ...
clvolkov's user avatar
  • 323
3 votes
0 answers
83 views

Transformation of Julia set sequence emerging from meromorphic function

I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form $$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$ ...
Jens Fischer's user avatar
0 votes
0 answers
72 views

Meromorphic extension of a limit function

Suppose $f_j(z)$, $j=1,2,..$ is a sequence of meromorphic functions on the complex plane $\mathbb{C}$. With a common set of all poles given by $S = \{-i,-2i, -3i,..\}$. Assume that each of them is ...
2inftyandBeyond's user avatar
3 votes
0 answers
163 views

Topology of level sets for meromorphic function

Let $F$ be a meromorphic function on $\mathbb{C}$. I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ ...
kaleidoscop's user avatar
  • 1,298
2 votes
1 answer
326 views

Lindelöf paper on meromorphic singularities

Does anyone know a digital link to the following paper, written by Ernst Lindelöf: "Mémoire sur certaines inégalités dans la théorie des fonctions monogènes, et sur quelques propriétés nouvelles ...
Felixson's user avatar
  • 232
0 votes
1 answer
153 views

Solutions of complex linear difference equations

I'm wondering what the solutions of complex linear difference equations like \begin{equation} f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C} \end{equation} look ...
Dawn's user avatar
  • 3
3 votes
1 answer
260 views

Meromorphic function on the Riemann surfaces

Let $V$ be a Riemann surface, $x\in V$, and $B:=B(x,r)$ some small ball (in a local chart). It is well known that there is a meromorphic function $f$ on $V$ with the only pole at $x$. What I’d like to ...
Lukasz Kosinski's user avatar
2 votes
1 answer
164 views

On a rigidity question related to spherical derivative of meromorphic functions

The spherical derivative of a meromorphic function is defined as $$f^\#(z):=\frac{|2f'(z)|}{1+|f(z)|^2}.$$ The motivation is that given a piecewise smooth curve $\gamma$ in the complex plane, the ...
student's user avatar
  • 1,330
0 votes
1 answer
283 views

Can a doubly periodic function be locally univalent?

I am looking for a meromorphic doubly periodic function such that the function is locally univalent. A standard meromorphic doubly periodic funtion is the Weirestrass $\wp$ function, defined as $$\wp(...
student's user avatar
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Does a vector over the field of meromorphic functions describe a manifold?

Assume that the variables $\mathbf x=(x_1,...,x_n)$ are coordinates on the solution manifold of a differential equation $\mathbf D(\mathbf x,\dot{\mathbf x},\ldots,\mathbf x^{(\alpha)})=\mathbf 0$ ...
emma's user avatar
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1 vote
0 answers
90 views

Finite groups of meromorphic functions [closed]

Which finite groups are isomorphic to groups of meromorphic functions on the whole complex plane under composition?
Daniel Sebald's user avatar
6 votes
1 answer
568 views

Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?

Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via $$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
Ali's user avatar
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4 votes
1 answer
544 views

Relationship between Dolbeault and de Rham cohomology on Riemann surface

A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $...
Martin Skilleter's user avatar
2 votes
0 answers
318 views

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 (...
John117's user avatar
  • 395
3 votes
1 answer
266 views

weakly holomorphic modular forms with a simple pole at $\infty$

Let $l$ be a prime. Suppose that $M_0^{!}(\Gamma_0(l))$ donote the space of weakly holomorphic modular forms of weight $0$ for the congruence subgroup $\Gamma_0(l)$. Does there exist a $f\in M_0^{!}(\...
W. Ma's user avatar
  • 33
2 votes
0 answers
128 views

principal divisor on complex surfaces

Let $X$ be a non compact complex surface non projective and non algebraic, and let $S$ be compact Riemann surface embedded in $X$ ( i mean that $S$ is a compact complex sub variety of $X$ of ...
singularity's user avatar
4 votes
1 answer
469 views

Rouché's Theorem in complex analysis on the relation of the number of zeros and poles of meromorphic functions in a region [closed]

This question is from my son referenced in my earlier question, Need advice or assistance for son who is in prison. His interest is scattering theory . He asked me to post this question: Hello and ...
Dan Cunningham's user avatar
0 votes
1 answer
343 views

Proof of the analytic Fredholm theorem in Borthwick

I've stumbled across a proof of the analytic Fredholm theorem given in Theorem 6.1 in Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick (see below). Given the notion of being &...
0xbadf00d's user avatar
  • 181
7 votes
1 answer
313 views

Indeterminacy locus of meromorphic maps of rigid analytic spaces

Setup. Let $k$ be an algebraically closed field of characteristic zero. Let $X/k$ be a normal variety, and let $Y/k$ be a proper variety. It is well-known that the indeterminacy locus of a rational ...
Jackson Morrow's user avatar
0 votes
1 answer
110 views

Is there any non-normal family $\mathcal{F}$ of meromorphic functions on $|z|<1$ whose each zero has multiplicity $2$ but $\mathcal{F'}$ is normal

It is well known that if a family of meromorphic functions is not normal (a family is said to be normal if each sequence of functions in the family has a subsequence which converges locally uniformly ...
Nik's user avatar
  • 165
1 vote
0 answers
427 views

Dimension of global holomorphic sections of a line bundle

Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
Roch's user avatar
  • 35
2 votes
1 answer
257 views

What is meromorphic differentials like on Riemann Sphere? [closed]

There is a proposition that every meromorphic differential on Riemann Sphere (or $\mathbb{P}^1 = \mathbb{C} \cup \{ \infty \}$) can be written as $f dz$ where $f$ is a meromorphic function on $\mathbb{...
bojohnzhang's user avatar
1 vote
1 answer
217 views

Some simple algebra of rational functions by André Weil

In André Weil's dissertation, he considers two meromorphic functions $x,y$ on a complex curve.  He assumes every pole of $y$ is a pole of $x$, and its multiplicity as a pole of $y$  is no greater than ...
Colin McLarty's user avatar
1 vote
0 answers
195 views

On a map between Riemann surfaces of genus $1$

Let $C$ be a compact Riemann surface of genus $1$, and $p\in C$, and $w$ be a local holomorphic coordinate on $C$ near $p$ with $w=0$ at $p$. As usual, for a divisor $D$ denote by $L(D)$ the vector ...
DesmondMiles13's user avatar
0 votes
0 answers
87 views

Coefficients of a special meromorphic function

The problem described below appears elementary, but I can't figure out the answer or find it in the literature. I apologize if I have missed something very basic. Let me begin with considering a ...
Dierk Bormann's user avatar
4 votes
1 answer
222 views

Poles of an integral of a meromorphic function with toric poles

Suppose I have a meromorphic function in several variables $f(x_1,\ldots,x_k,y_1,\ldots,y_m)$ and I want to integrate along the torus $T^m$ given by $|y_1|=\cdots=|y_m|=1$. It is not true in general ...
Anton Mellit's user avatar
  • 3,592
5 votes
1 answer
260 views

A "prequestion" about meromorphic representations of algebraic groups

In a comment exchange around an answer to Is a group scheme determined by its category of representations? there arose the issue of Tannakian reconstruction for non-affine algebraic groups (e. g. ...
მამუკა ჯიბლაძე's user avatar
3 votes
0 answers
121 views

Tilings of the plane and meromorphic functions on the plane

This question has three up-votes on m.s.e. but isn't getting any answers. Every textbook says every doubly-periodic meromorphic function on $\mathbb C$ has a fundamental domain that is a ...
Michael Hardy's user avatar
2 votes
1 answer
1k views

When may function (meromorphic) be expanded as power series with coefficients of integers

Let $F$ be meromorphic function, with what properties may it be expanded as power series with coefficients of integers in such a form: $$F=\sum_0^{\infty}a_i x^i,a_i\in \mathbb{N} \bigcup 0,\exists M \...
XL _At_Here_There's user avatar