Questions tagged [meromorphic-functions]
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34 questions
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Weak Phragmén-Lindelöf for meromorphic functions?
Let $f$ be meromorphic on the half-strip $\Re s\leq 0$, $|\Im s|\leq 1$, and bounded on its boundary. I would like to be able to assert that there is a sequence $0>\sigma_1>\sigma_2>\sigma_3&...
- 22k
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145
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Application of Hiromi's & Ozawa's lemma by Rubel & Yang
The paper by Hiromi & Ozawa proves the following lemma:
Let $a_0(z), a_1(z), \dots, a_n(z)$ be meromorphic functions and $g_1(z), \dots, g_n(z)$ entire functions. If
$$
T(r,a_j) = o \left( \sum_{v=...
- 11
6
votes
1
answer
625
views
Meromorphic functions bounded from below and above on a discrete set such as $\mathbb Z + i \mathbb Z$
Let $f$ and $g$ be entire functions and let $D \subset \mathbb C$ be a discrete set, for example $D=\mathbb Z + i \mathbb Z$. Suppose that the meromorphic function $h=\frac{f}{g}$ is bounded on $D$ ...
- 66
9
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0
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532
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Reframing Collatz Conjecture as a property of meromorphic functions
I was wondering if it is known that the 3n+1 Collatz conjecture could be reframed as a statement about the set of solutions to a particular equation formulated as the sum of residues. This is ...
- 107
2
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1
answer
147
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Representation of a meromorphic function on a once-punctured complex plane in terms of its zeros and poles
Consider a meromorphic function $f:\mathbb{C}\setminus\{0\}$ such that both $0$ and $\infty$ are its essential singularities with finite order in the sense of value distribution theory (see for ...
- 63
2
votes
2
answers
332
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Behavior of the hypergeometric function near x=1
It is known that the hypergeometric function ${}_2F_1(a, b, c; x)$ defined by the series
$$\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$$
behaves ...
- 393
3
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0
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127
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Transformation of Julia set sequence emerging from meromorphic function
I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form
$$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$
...
- 377
0
votes
0
answers
115
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Meromorphic extension of a limit function
Suppose $f_j(z)$, $j=1,2,..$ is a sequence of meromorphic functions on the complex plane $\mathbb{C}$. With a common set of all poles given by $S = \{-i,-2i, -3i,..\}$.
Assume that each of them is ...
3
votes
0
answers
237
views
Topology of level sets for meromorphic function
Let $F$ be a meromorphic function on $\mathbb{C}$.
I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ ...
- 1,362
2
votes
1
answer
387
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Lindelöf paper on meromorphic singularities
Does anyone know a digital link to the following paper, written by Ernst Lindelöf:
"Mémoire sur certaines inégalités dans la théorie des fonctions monogènes,
et sur quelques propriétés nouvelles ...
- 1
0
votes
1
answer
220
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Solutions of complex linear difference equations
I'm wondering what the solutions of complex linear difference equations like
\begin{equation}
f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C}
\end{equation}
look ...
- 3
3
votes
1
answer
400
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Meromorphic function on the Riemann surfaces
Let $V$ be a Riemann surface, $x\in V$, and $B:=B(x,r)$ some small ball (in a local chart). It is well known that there is a meromorphic function $f$ on $V$ with the only pole at $x$. What I’d like to ...
2
votes
1
answer
266
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On a rigidity question related to spherical derivative of meromorphic functions
The spherical derivative of a meromorphic function is defined as
$$f^\#(z):=\frac{|2f'(z)|}{1+|f(z)|^2}.$$
The motivation is that given a piecewise smooth curve $\gamma$ in the complex plane, the ...
- 1,360
0
votes
1
answer
333
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Can a doubly periodic function be locally univalent?
I am looking for a meromorphic doubly periodic function such that the function is locally univalent.
A standard meromorphic doubly periodic funtion is the Weirestrass $\wp$ function, defined as
$$\wp(...
- 1,360
0
votes
0
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82
views
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 $\mathbf D(\mathbf x,\dot{\mathbf x},\ldots,\mathbf x^{(\alpha)})=\mathbf 0$ ...
- 131
1
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0
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102
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Finite groups of meromorphic functions [closed]
Which finite groups are isomorphic to groups of meromorphic functions on the whole complex plane under composition?
- 3,213
6
votes
1
answer
601
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+\...
- 4,189
4
votes
1
answer
738
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 $...
- 811
2
votes
0
answers
424
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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 (...
- 435
3
votes
1
answer
333
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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^{!}(\...
- 133
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156
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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 ...
- 135
3
votes
1
answer
662
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 ...
- 543
0
votes
1
answer
506
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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 &...
- 249
7
votes
1
answer
382
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 ...
- 1,008
0
votes
1
answer
163
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 ...
- 165
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542
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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 ...
- 35
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1
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357
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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{...
- 49
1
vote
1
answer
239
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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 ...
- 11.4k
1
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265
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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 ...
0
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0
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108
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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 ...
- 686
4
votes
1
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278
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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 ...
- 3,832
5
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1
answer
298
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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. ...
- 19.4k
3
votes
0
answers
173
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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 ...
- 12.9k
4
votes
1
answer
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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} \cup \{0\},\exists M ...
- 3,316