# 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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### 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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### 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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### 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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### 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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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, ...