Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
0
votes
0answers
13 views
An integral involving the argument of the Gamma function
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 ...
2
votes
0answers
89 views
Prove conjugation law for Exp(z) using only Exp(x + y) = Exp(x)Exp(y)
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
22 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
78 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 ...
-5
votes
0answers
146 views
On the zeros of the Riemann zeta function [on hold]
Let $t$ denote the imaginary part of a zero $\rho$ of the Riemann zeta function with $\Re(\rho)>\theta,$ where $1/2 \leq \theta<1$. Let $f$ be some other function, not identically $0$. Suppose ...
18
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
151 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(...
0
votes
0answers
73 views
How to understand the sign periodicity of any conditionally convergent series? [closed]
Given any conditionally convergent series $\sum_{n\geq1} a_n$. I wonder if there is a "standard way/method" to "investigate/estimate" the sign periodicity of $a_n$ by some explicit functions $f(x)$ i....
1
vote
0answers
62 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
74 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
274 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
156 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
89 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
43 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
45 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
94 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
336 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
147 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 ...
0
votes
0answers
39 views
For $|q|<1$, the function $\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on $|z|<1$
I want to prove that for $|q|<1$, the function $f(z):=\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on the set $\{z:|z|<1\}$.
My approach: We consider the sequence of functions $\{f_n\}$ ...
7
votes
1answer
175 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
68 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
220 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)|$$.
where $\zeta$ ...
2
votes
3answers
205 views
Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$
Denote by $\zeta$ the Riemann zeta function.
It is known that
$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$
But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
2
votes
1answer
168 views
Exponential type of a product of entire functions
Let $\{a_n\}_{n=1}^\infty$ and $\{b_m\}_{m=1}^\infty$ be two sequences of points in $\mathbb{C}$ such that
$$
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
37 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
58 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:
$$ \frac{\theta^...
2
votes
0answers
80 views
Understanding proof of q-theta function expression
In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ 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
201 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 ...
-1
votes
0answers
128 views
When a potential is real analytic
Suppose $B$ is the ball of center $a$ and radius $R>0$ in $ \mathbb{R}^{n} $ $n>1$. Suppose also that $u$ is subharmonic and real analytic on a neighborhood of the closure of $B$. We know that ...
2
votes
1answer
78 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
139 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
48 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
$$ H(t)= H(...
2
votes
0answers
31 views
Smoothings over a real interval
I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear.
Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...
2
votes
1answer
79 views
Smoothings of isolated non-irreducible surface singularities
Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
4
votes
1answer
127 views
Julia set containing smooth curve
I have two realted questions.
Let $R$ be a rational function on $\mathbb{C}$ with degree at least 2. We denote by $\mu$ the measure of maximal entropy for $R$ and recall that the Julia set coincides ...
1
vote
1answer
100 views
Computing the convex hull of a region of $\mathbb{C}^2$
Consider a function $f(z, w)$ of two complex variables. The function is symmetric with respect to $z$ and $w$. When $\Re(z)>0$ and $\Re(w)>0$, the function is analytic in its two variables. When ...
4
votes
0answers
46 views
Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains
Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...
2
votes
0answers
63 views
The use of concavity of $\log\det\left(u_{j\overline{k}}\right)$
Let $F:\mathbb{C}^{n\times n}\rightarrow\mathbb{C}$ be the function
$$
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
108 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
60 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 $\mu$...
14
votes
3answers
609 views
How to find a conformal map of the unit disk on a given simply-connected domain
By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
2
votes
0answers
153 views
Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
3
votes
0answers
57 views
Is a domain of a holomorphic flow pseudoconvex?
Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
2
votes
0answers
179 views
Does the Riemann Xi function possess the universality property?
Here is the question.
Does the Riemann Xi function possess the universality property, or something similar to Voronin 's universality property?
This is the reason why the answer to this question ...
3
votes
1answer
105 views
A question about distribution of fractional part of $2^k\alpha$
Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$.
The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...
2
votes
1answer
238 views
Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?
Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
6
votes
1answer
253 views
Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$
Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.)
Is it true that for any ...
8
votes
0answers
78 views
Borel-Ecalle re-summation and resurgence: Criteria and Results
This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\...
