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.
319 questions
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An $\operatorname{SL}_2(\mathbb{Z})$-invariant meromorphic function with large poles
I am learning about modular forms and modular functions, and in working out the nitty-gritty details, I have stumbled on the following question:
Is it possible to find an $\operatorname{SL}_2(\mathbb{...
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Doubts on a proof of Mittag-Leffler theorem
I am having a hard time understanding the proof of Mittag-Leffler theorem as a consequence of Runge's theorem in the book "Complex Made Simple" by David Ullrich. The first part is similar to ...
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How to extend a function in real axie to a meromorphic function in complex plane? [duplicate]
Background. In physics, experimental observables are often related to real-time or real-frequency Green’s functions, $G(t)$ or $G(\omega)$, which are (typically) real-analytic. Physicists often extend ...
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Meromorphic symbol pseudodifferential operator
Context: For some function $f\in\mathcal{M}(\mathbb C)$ (meromorphic function on $\mathbb C$), I am interested in linear operators $T_f$ that act on functions of the form $g_a:x\mapsto \exp(ax)$ in ...
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Help with an inequality regarding the chordal metric
I am going through my professor's notes for complex analysis and a given inequality is driving me up the wall.
It is used as part of the proof that states that if a sequence of meromorphic functions ...
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Prove that a $p$-adic function is meromorphic
In notes of a lecture, I found this exercice:
let $p$ be a prime number. Show the function $f$ defined on $\mathcal D=D(0,1)=\{z\in\mathbb C_p\mid|z|_p<1\}$ by $\displaystyle f(z)=\sum_{n=0}^{+\...
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Existence of a certain pair of meromorphic maps
Let $U$ be an open subset of $\mathbb{C}$. Can one find two holomorphic (or maybe better phrased here - conformal) inclusions $f, g: U \rightarrow \mathbb{CP}^1$ such that the distance from $f(z)$ to $...
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What is a nice class of maps from a real manifold to a complex manifold which satisfies a maximum principle?
Let $X$ be a smooth real manifold and let $Y$ be a complex manifold. I would like to know what are some known classes of maps from $X$ to $Y$, such that if $f: X \to Y$ is any map in that class, then ...
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Determining all holomorphic functions that conditions on the singularities
I came across a complex analysis exercise which I was not able to figure out.
Let $\Omega = \mathbb{C} \setminus \{ i - 1 , i + 1 , 2i \}$. Determine all functions $g \in H(\Omega)$ such that
$(i)$ $\...
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Reference on existence of meromorphic square root
Context: I am searching for a reference for the following result:
Let $f:U\to \mathbb C$ be a meromorphic function. There exists a meromormphic function $g:U\to \mathbb C$ such that $g^2=f$ if and ...
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When are Hurwitz Zeta's polynomials of each other?
For $z\in \mathbb{C}$, let $$e_1(z) = \frac 1 z + \sum_{n\geq 1} \left(\frac 1 {z+n} + \frac 1 {z-n}\right)$$
and for $k\geq 2$,
$$e_k(z) = \sum_{n\in \mathbb{Z}} \frac 1 {(z+n)^k}.$$
These $e_k$ are ...
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Meromorphic function $g$ sharing value(s) with its derivative $g'$ counting multiplicity (CM). What does it mean?
I am working on some of the fundamental results of functions sharing values with their derivatives as an application of Nevanlinna Theory. There are many results for meromorphic functions sharing ...
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How to construct all functions on the torus
I was reading section 3.2.3 from riemannian surfaces by the way of complex analytic geometry,
Now we will construct all functions on the torus. We specialize to a lattice of the form $L=\mathbb Z+\...
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Meromorphic function takes each value of the Riemann sphere the same number of times
Let $f$ be a nonconstant meromorphic function on the Riemann sphere $\hat{\mathbb{C}}$. How do I show that $f$ takes each value in $\hat{\mathbb{C}}$ the same number of times (counting multiplicity)?
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About meromorphic 1-forms on complex manifolds.
Suppose that $M$ is complex manifold of dimension $m\geq 2$. By definition, a holomorphic $1$-form $\omega$ on $M$ is a holomorphic section of the holomorphic cotangent bundle (which is an holomorphic ...
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