# 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.

293 questions
Filter by
Sorted by
Tagged with
1 vote
26 views

### 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 "...
• 552
1 vote
38 views

### 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 ...
• 31
123 views

### 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'...
1 vote
60 views

• 177
20 views

• 459
149 views

### 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 $\begin{pmatrix} a&b\\ c&d \end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$ ...
• 420
147 views

### 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 ...
• 500
1 vote
39 views

### 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 ...
• 1,901
33 views

### 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 $1>r>0$,...
• 5,531
1 vote
32 views

### 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 ...
• 16k
38 views

### 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 ...
• 319
57 views

### 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 ...
• 1
202 views

• 1,786
23 views

### 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 $\bi_n$...
### 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 ...