Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
2
votes
0answers
91 views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
2
votes
0answers
31 views
Entire analytic functions with entire analytic Fourier transform, and corresponding distributions
I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...
0
votes
0answers
18 views
Parametric statistics: how to estimate the supremum of a set of parameters from a random sample
I would like to ask a question on how to estimate the supremum norm of a set of parameters in the following setting. I appreciate any pointer or suggestion. Thanks.
Question:
Suppose we have $m$ ...
4
votes
2answers
135 views
Density of Lacunary Functions
I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ...
0
votes
1answer
60 views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
5
votes
2answers
139 views
Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
4
votes
1answer
126 views
Compilation of representations of holomorphic functions
Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse.
A holomorphic function $f$ on the unit disk $\...
2
votes
1answer
95 views
Extended Abel-Jacobi theorem
Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...
3
votes
2answers
251 views
On finite extensions of the field of meromorphic functions
Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation
$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + ...
-1
votes
1answer
197 views
On a certain representation of the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
4
votes
0answers
70 views
LlogL and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
1
vote
1answer
542 views
An integral involving the argument of the Gamma function and the Riemann Hypothesis
Evaluate $$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
Note that $I$ converges ...
8
votes
1answer
219 views
Bounded holomorphic functions on a Riemann surface separating points
Let $R$ be a Riemann surface that admits a non-constant bounded holomorphic function. Then is it true that any two points of $R$ can be separated by a bounded holomorphic function? This is easy to see ...
2
votes
0answers
109 views
Prove conjugation law for Exp(z) using only Exp(x + y) = Exp(x)Exp(y) [closed]
Starting with just the property $E(x + y) = E(x)E(y)$, one can prove quite a lot of the main properties of the exponential function on real numbers. For example, $E(0) = 1$, and $E'(X) = E(X)$, and $E(...
0
votes
0answers
32 views
Steepest descent integration in several dimensions
The method of steepest descent provides an asymptotic approximation for integrals of the form:
$$I = \int_C \exp(M f(z))\mathrm dz$$
for large positive $M$, where $f(z)$ is analytic in the region of ...
5
votes
1answer
112 views
Unconditionaly convergent series in some functional spaces
Linked with [this question and discussion](
Bilinear product of two summable families), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall ...
19
votes
2answers
1k views
Complex analytic vs algebraic geometry
This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next.
It looks to me, that complex-analytic geometry has lost its relative positions ...
1
vote
1answer
167 views
Inquiry on the bound for $\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$
Let $\zeta$ be the zeta function of Riemann. Is the bound for
$$I_{T}=\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$$
known ?
It seems to me that $I_{T} \ll T\log T$ since $\log|\zeta(...
1
vote
0answers
68 views
Is the set of fixed points of hyperbolic elements from $\mathrm{PSL}_{2}(A)$ a group?
Given a subring $A$ of $\mathbb{R}$, we can consider the set $\mathrm{PSL}_{2}(A)$ of elements in $\mathrm{PSL}_{2}(\mathbb{R})$ with entries in $A$ and the determinant of associated matrix is equal ...
0
votes
0answers
82 views
Upper bound of $\zeta$-function on critical strip
How can I determine any upper bound for $|\frac{\zeta^4(s)}{\zeta(2s)}+d\cdot\zeta^2(s)|$ on the critical strip $s=\frac12+it$ for an integer $d$?
5
votes
1answer
311 views
On limits of manifolds
This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...
4
votes
0answers
164 views
Computing Bohr Radii
The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\...
3
votes
1answer
93 views
The extension of a plurisubharmonic Functions
I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by Phillip A. Griffiths. Proposition 2.9 of the paper is: If $\Psi$ is a plurisubharmonic on the punctured ball $B_n^{*}$ ...
1
vote
1answer
45 views
On extensions of holomorphic mappings with image in a projective algebraic variety
I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that:
Let $N$ be a complex manifold, $S\...
1
vote
0answers
48 views
How to fix multi-valued function on contour?
I am sorry to ask such an embarrassingly simple question here. My question is about contour integral of the multivalude function.
I want to calculate the Fourier transformation of a muti-valued ...
0
votes
1answer
126 views
Poles of equivariant meromorphic functions on Riemann surfaces
Let $p:\Sigma\to \mathbb{P}^1$ be the cyclic cover of $\mathbb{P}^1$ with Galois group $\Gamma$. Let $\Gamma\cdot p$ be a free $\Gamma$-orbit on $\Sigma$. Given any character $\chi$ of $\Gamma$, does ...
16
votes
1answer
358 views
Cohomology of real analytic coherent sheaves
Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...
3
votes
1answer
152 views
Is a factorial scheme with Noetherian stalks locally Noetherian?
Motivation: The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct ...
7
votes
1answer
176 views
Cauchy path integral as a linear operator: kernel and image?
Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
1
vote
0answers
72 views
Meromorphic mappings between complex projective spaces
Let $n>2$ and $\phi: \mathbb{P}_{\mathbb{C}}^n \setminus S \rightarrow \mathbb{P}_{\mathbb{C}}^n$ be a holomorphic map and $S$ a closed analytic subset of $\mathbb{P}_{\mathbb{C}}^n$ with ...
-2
votes
1answer
230 views
A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$
By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that
$$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$.
where $\zeta$ ...
2
votes
3answers
209 views
Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$
Denote by $\zeta$ the Riemann zeta function.
It is known that
$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$
But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
2
votes
1answer
170 views
Exponential type of a product of entire functions
Let $\{a_n\}_{n=1}^\infty$ and $\{b_m\}_{m=1}^\infty$ be two sequences of points in $\mathbb{C}$ such that
$$
f(z)=\prod_{n=1}^\infty\left(1-\frac{z}{a_n}\right)\quad\mbox{and}\quad g(z)=\prod_{m=1}^\...
0
votes
0answers
38 views
Simplifying a general complex function
Consider the follwing complex function:
$\frac{1}{1+ \lambda f(z)f(\frac{1}{z})},$
where $\lambda$ is a real constant number and $z$ is a complex variable.
I need to find a function $G(z)$ such ...
0
votes
0answers
62 views
Laurent series expansion of Theta function expression
Using the product definition of the theta function
$$ \theta(z;q) = \prod_{k=0}^{\infty}(1-q^k x)(1-q^{k+1}/x) $$
I would like to find the Laurent series expansion of the following:
$$ \frac{\theta^...
2
votes
0answers
81 views
Understanding proof of q-theta function expression
In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...
3
votes
2answers
215 views
Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability
We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
3
votes
1answer
103 views
Variation of steepest descent/Laplace methods for non-exponential integrands
I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type
$$\int_C f(z) M(\lambda g(z)) dz$$
for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
0
votes
1answer
140 views
Complex structures on topological surfaces
I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex ...
2
votes
0answers
49 views
Equations needed to define a normal complex surface singularity
This questions is highly related with this other question of mine: Irreducible surface singularity that is not a local set-theoretical complete intersection I just thought that a different look at the ...
0
votes
0answers
41 views
What can we say about the Bargmann transform of bounded function?
Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$
Now we define
$$ H(t)= H(...
2
votes
0answers
31 views
Smoothings over a real interval
I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear.
Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...
2
votes
1answer
79 views
Smoothings of isolated non-irreducible surface singularities
Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
4
votes
1answer
132 views
Julia set containing smooth curve
I have two realted questions.
Let $R$ be a rational function on $\mathbb{C}$ with degree at least 2. We denote by $\mu$ the measure of maximal entropy for $R$ and recall that the Julia set coincides ...
1
vote
1answer
109 views
Computing the convex hull of a region of $\mathbb{C}^2$
Consider a function $f(z, w)$ of two complex variables. The function is symmetric with respect to $z$ and $w$. When $\Re(z)>0$ and $\Re(w)>0$, the function is analytic in its two variables. When ...
4
votes
0answers
46 views
Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains
Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...
2
votes
0answers
63 views
The use of concavity of $\log\det\left(u_{j\overline{k}}\right)$
Let $F:\mathbb{C}^{n\times n}\rightarrow\mathbb{C}$ be the function
$$
F\left(a_{1\overline{1}},a_{1\overline{2}},\ldots,a_{n\overline{n}}\right):=\log\det\left(\begin{array}{ccc}
a_{1\overline{1}} &...
4
votes
1answer
114 views
Analytic continuation of 2 variable function
Consider a function $F(x, y)$ of two complex variables. For $\Re(y)>0$, we know the analytic structure of the function. In that case, the function is meromorphic, with simple poles in $x$ at ...
1
vote
0answers
53 views
Global interior estimate complex Monge-Ampere equation
Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...
1
vote
1answer
62 views
Riesz measure of a smooth subharmonic function on a ball
Let $B(x_{0},r)$ be the ball of center $ x_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x_{0},r)$. Let $\mu$...
