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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Meromorphic continuation of $\zeta(s)$ to ${\rm Re}(s)>0$.
Riemann zeta function is one of the most mysterious functions that we encounter in mathematics.We require a meromorphic continuation of this function to ${\rm Re}(s)>0$ in order to prove the prime ...
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Definition of meromorphic function between complex manifolds
Ususally we only consider meromorphic function from a complex manifold $X$ to $\mathbb{C}$:
Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is sheaf of holomorphic functions. We ...
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Meromorphic functions on a compact Riemann surface as a "discrete" Lie group
Let $X$ be a compact Riemann surface and let $\mathscr{M}^\ast(X)$ be the set of non-zero meromorphic functions on $X$. Clearly $\mathscr{M}^\ast(X)$ is a multiplicative abelian group.
Let $\mathscr{D}...
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What is the definition of sheaf of meromorphic differential form on a complex manifold？
Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is the sheaf of $\mathbb{C}$-valued holomorphic functions on $X$.
Let $T_X^\vee$ be the cotangent bundle over $X$ with the ...
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Show that $\frac{f'(z)}{f(z)}$ has a pole if and only if $f(z)$ has a zero or a pole.
In our analytic number theory course, our instructor told us that $\zeta(s)$ has a zero or a pole at $z_0$ if and only if $\frac{\zeta'(s)}{\zeta(s)}$ has a pole at $z_0$. In fact, this is true for ...
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conformal mapping from unit disk to equilateral triangle
Let $$f(z) = C \int_{0}^{z} \frac{1}{(1-\zeta^3)^{\frac{2}{3}}}d\zeta$$ with $$C \int_{0}^{1} \frac{1}{(1-x^3)^{\frac{2}{3}}}dx=1$$ be a conformal map from the unit disc $B(0,1)$ to the equilateral ...
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Power series expansion of 1/(1+z)^2 around z=1 [duplicate]
I'm struggling with getting the power series/Taylor series expansion of $f(z) = \frac{1}{(1+z)^2}$ around $z_0 = 1$.
Usually, I would do a partial fraction decomposition, and then do some re-arranging ...
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Solutions of complex linear difference equations
I know solutions of complex linear differential equations like
\begin{equation}
f^{(k+1)}(z)+c_{k-1} f^{(k)}(z)+\ldots+c_1 f^{\prime}(z)+c_0=0
\end{equation}
are linear combinations of $e^{\alpha z}$. ...
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Riemann-Roch on a hyperelliptic Riemann surface
Let $X$ be a hyperelliptic compact Riemann surface of genus $g\geq 1$. This means that there are two distinct points $p,q\in X$ such that $\ell(p+q)=2$. I'm trying to prove that:
$$\ell((g+1)(p+q))=g+...
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Exercise on Riemann surfaces
Let $X$ be a compact Riemann surface of genus $g\geq 2$ and let $p,q\in X$ be two distinct points. I want to prove that:
$$\ell(p+q)\in \{1,2\}$$
Clearly $\ell(p+q)\geq 1$ because it contains ...
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Equality between $\text{gcd}$ and $\text{lcm}$ of divisors on a compact Riemann surface
Notation: If $E$ is a compact divisor on a Riemann surface and $p\in X$, I'll use the notation $E^{(p)}$ to denote the coefficient of the point $p$ in $E$.
Let $D\geq 0$ be a divisor on a compact ...
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Difficulties in finding the radius of the disk in which power series of a function is analytic
I am trying to find a function of asymptotic growth of the Fibonacci sequence, which's generating function is
$$ f(z)=\frac{1}{1-z-z^2}=\frac{1}{(1-\phi _-z)(1-\phi_+z)}
$$
with poles at
$
z_1=-\phi_- ...
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Kernel of $\text{Res}_p$ on a Riemann surface
Let $X$ be a Riemann surface and let $p\in X$. Let:
$$\text{Res}_p:\Omega^1_{\text{mero}}(X)\to \mathbb{C}$$
be the residue at $p$ operator defined on meromorphic $1$-forms. Clearly holomorphic $1$-...
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Meromorphic function - Countably infinite or finite number of poles?
I am having trouble reconciling the following two questions:
Can it be proved that a Meromorphic function only has a countable number of poles?
$f$ meromorphic on $\mathbb{\hat{C}}$ $\implies$ $f$ ...
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Schwarz reflection principle for meromorphic functions
I'm trying to work on this exercise from Conway's Complex Analysis textbook (Exercise 3, Chapter 9, Section 1). Here $G$ denotes a symmetric region, $G_+$ the points of $G$ above the real axis and $...
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Why is $\mathcal K_X$ the "sheaf of rational functions"?
Let $f:X\to\mathbb A$ be a regular function on a scheme. Is there some relation between $f$ being a non-zero divisor in $\mathcal O(X)$ and the denseness (topologically or scheme theoretically) of $D(...
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Can I use Bohr-Mollerup Theorem to prove Legendre duplication formula?
CONTEXT
The proof of Legendre duplication formula is required as an exercise in Conway's "Functions of One Complex Variable I".
I wanted to base my proof only on the material presented up to ...
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Confusion on a note by Conway on Marty's Theorem
I am stuck on a note by Conway right before the proof of Marty's Theorem (equivalent of Montel's Theorem, for meromorphic functions) in "Function of One Complex Variable I".
The family of ...
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How to prove the quotient of homogenous polynomial with same degree is a meromorphic function on $\Bbb{CP}^n$
Let $f,g$ be two homogenous polynomial of degree $d$ in $\Bbb{C}[x_1,...,x_{n+1}]$. define the quotient $f/g$ on $\Bbb{CP}^n$, intuitively it should be the meromorphic function, but I want to prove it ...
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Order of vanishing along the hypersurface [duplicate]
I was reading Huybrechts complex geometry textbook, the definition of order of vanishing for the meromorphic function along the hupersurface is as follows:
Let $x\in Y$ where $Y$ is a hypersurface in ...
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Definition of meromorphic function on complex manifolds
There are typically four(or more?) definition of meromorphic function on Riemann surface and three definitions of meromorphic function on complex manifolds, I want to show they are equivalent (I will ...
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Analytic function and derivatives
Show that a smooth complex function $f(z, \bar{z})$ satisfies $\frac{\partial^2 f}{\partial \bar z^{2}}=0$ if and only if $f(z, \bar{z})=\bar{z}g(z)+h(z)$ for some analytic functions $g(z)$ and $h(z)$....
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Show that $\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$ is meromorphic
Let $p\geq 1$ be an integer. If $\sum_{n=1}^\infty 1/|a_n|^{p+1}$ converges, show that
$$\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$$
is ...
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Inverse and extension of the $n$th Airy Ai and Bi zero
$\def\ai{\operatorname{ai}} \def\bi{\operatorname{bi}} \def\Ai{\operatorname{Ai}} \def\Bi{\operatorname{Bi}} $ Airy Ai Zero $\ai_n$ gives the $n$th zero of the Airy Ai function and Airy Bi Zero $\bi_n$...
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Can a meromorphic function $f:U \to \mathbb{C}, \; U\subset \mathbb{C}$ domain, that is not the zero function, have a zero of order infinity?
Can a meromorphic function $f:U \to \mathbb{C}, \; U\subset \mathbb{C}$ domain, that is not the zero function, have a zero of order infinity?
(Clear: Would $f$ be holomorphic on the whole domain U ...
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Existence of a meromorphic orthogonal matrix
Is there any meromorphic matrix $F$ on $\mathbb{C}^*$, with a non trivial pole at $0$ and with values in $O_2(\mathbb{C})$ ?
My guess is that a such matrix doesn't exist, but i'm not sure.
I've tried ...
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Meromorphic function on extended complex plane is open
I have seen an answer of this question on the link A meromorphic function is open?
I understand that these are the cases but I am not getting how to elaborate for example if $z\in G$ and $f(z)=\infty$ ...
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theorem of Mittag-Leffler
I want to find a meromorphic function that has poles exactly in the natural numbers and has the principal part $\frac{1}{z-n}.$
Therefore I used the theorem of Mittag-Leffler
and I received $\sum_{n=1}...
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Sameness of Riemann surfaces
I need to show that the equations $x^3+y^3=1$ and $y^2=4x^3-1$ are the same Riemann surface in $\mathbb{CP}^2$ and as a consequence to show that there two meromorphic functions $f,g$ such that $f^3+g^...
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$\int_{|z|=n} z^m\tan z\;dz$
Evaluate$$\int_{|z|=n} z^m\tan z\;dz$$
I am solving this example 17 about generalized argument formula in a book
Now i have some doubts
Author says singularities of f(z) are $(k+{1\over2})\pi$ . I ...
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How to prove Theorem 4.10 (the Branched Covering Principle in the extended complex plane) from Bruce Palka's complex analysis textbook (Ex. 5.83)?
The following is Theorem 4.10 ("Branched Covering Prnciple") on p. 361 of Bruce P. Palka's An Introduction to Complex Function Theory, Springer, 1991 (corrected second printing of 1995).
...
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Meromorphic function at infinity
I'm studing about meremorphic functions in complex analysis and I'm trying to understand the proofe of being a mereomorphic function on $\mathbb{P}^1(\mathbb{C})$ means the function is rational. I ...
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coefficient of poles of meromorphic function on a Riemann surface does not depend on the local coordinate
Suppose $M$ is a Riemann surface and $f$ is a meromorphic function on $M$ who has a pole at $p$ of degree $n$. If $w,z$ are both local coordinates around $p$ such that $w(0)=z(0)=p$, then $f$ has a ...
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Prove that even doubly periodic function satisfies a differential equation
Let $f(z)$ be analytic in $\mathbb{C}\setminus\{m+ni:m,n\in\mathbb{Z}\}$. Assume that $f(z)=f(-z)$ , $f(z)=f(z+m+ni)$ and $f$ has a pole of order $2$ at $0$. Prove that there exist numbers $a_0,a_1,...
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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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Why are certain contour integrals of doubly periodic meromorphic functions zero?
Consider the following theorem: suppose $f:\mathbb C \to \mathbb C_{\infty}$ is a non-constant doubly periodic meromorphic function. (So, $f(z) = f(z+ \omega_1) = f(z+\omega_2)$ for all $z \in \mathbb ...
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How do I see that this function has removable singularities?
I have the following problem:
Let $f:\Omega \rightarrow \Bbb{C}$ be a meromorphic function in a region $\Omega$ with finately many zeros $z_j$ of order $m_j$ and finately many poles $p_k$ of order $...
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When does a Fuch's type 2nd order ODE not have a singularity at infinity?
I know that any second order linear ode
$$w''+p(z)w'+q(z)w=0$$
is of Fuchs type (ie, coefficients meromorphic with singular points at $z_0,...,z_n,\infty$ all regular) if
$$p(z)=\sum\frac{p_j}{z-z_j}, ...
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Meromorphic function which is analytic bounded on $\mathbb H$, has no zeroes on $\mathbb H^-$, modulus 1 on $\mathbb R$
I am interested in the class $G$ of meromorphic function $f:\mathbb C \to \mathbb C$ such that:
$f$ is analytic and bounded on the upper half plane $\mathbb H$.
$f$ has no zeroes on the lower half ...
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Meromorphic function with poles of unbounded order
Let:
$$
f(z) = \sum_{j=1}^{\infty} \frac{1}{(z - j)^{j}}
$$
so that $f$ has a pole of order $j$ at $z = j$ for each integer $j > 0$. Is $f$ meromorphic in the extended complex plane? It is analytic ...
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The quotient of homogeneous polynomial on a smooth projective curve is meromorphic if the denominator is not identically zero.
Suppose $X \subset \mathbb{CP}^n$ is a smooth projective curve.
Why is $G/H$ a meromorphic function on $X$ if $G$ and $H$ are homogeneous polynomials of degree $d$ and $H$ does not vanish identically ...
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To check sum of two function is meromorphic or not
$\phi(z) = Re(z) + f(z)
$
where $f(z)$ is a meromorphic function. then WOTF is true,
1)$\phi $ is meromorphic
2)$\phi $ and $ f(z) $ has same number of singularities
3)$\phi $ is analytics in every ...
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Meromorphic function inequality
I'm studying "Meromorphic Functions" by W. K. Hayman and I'm stuck with the proof for theorem 1.7.1 (p.18):
If $f$ is regular for $|z|\leqslant R$ and
$$M(r,f)=\max_{|z|=r}|f(z)|,$$
then
$$...
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Meromorphic functions in projective space
It is well known that meromorphic functions in $\mathbb P^n_\mathbb C$ are of the form $p/q$ where $p$, $q$ are homogeneous polynomials in $\mathbb C[x_0 , \ldots , x_n]$ of the same degree, and $q \...
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Power series with radius of convergence two.
Let $f(z)$ be a power-series (with complex coefficients) centered at $0 \in \mathbb{C}$ and with a radius of convergence 2 . Suppose that $f(0)=0$. Choose the correct statement(s) from below:\
$f^{-1}...
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Holomorphic function that has a triple pole at $0$, a simple pole at $1$, an essential singularity at $i$ and at $-i$
Find a holomorphic function that has a triple pole at $0$, a simple pole at $1$, an essential singularity at $i$ and at $-i$. I know that $f(z)=\frac{1}{z^3}+\frac1{z-1} + e^{\frac{1}{z^2+1}}$ is such ...
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Meromorphic function with non-negative real-part is in fact analytic/holomorphic
Let $f$ be a meromorphic function on an open set $\Omega$ satisfying $Re~f \ge 0$. Prove that $f$ is in fact holomorphic.
I'm not entirely sure how to solve this, but here are some thoughts. Let $z_0$...
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Possible values of $l(np)$ for $p\in X$, where $X$ is a compact Riemann surface of genus 2
Here $l(D)$ denotes the dimension of the complex vector space
$$\mathcal{L}(D)=\{f\colon X \to \mathbb{P}^1\mathbb{C} \hspace{5pt} \mathrm{meromorphic} \hspace{5pt} \mathrm{s.t.} \hspace{5pt} (f)+D \...
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Why is open mapping a geometric property?
I was recently reading Chapter 3 on meromorphic functions in "Complex Analysis by Stein and Shakarchi". They mentioned that the property that holomorphic functions are open mappings is a ...
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Are meromorphic functions generally (anti)conformal
All analytic functions are conformal whenever their derivatives aren’t $0$.
$x^{-1}$ is anticonformal (preserves angles but changes orientations) whenever its derivative isn’t $0$ or $\infty$
Are ...
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