Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
0
votes
0answers
45 views
Ratio of exponentially weighted Selberg integrals
I'm interested in bounding the following ratio of integral:
$$\frac{\int_{0<x_k<...<x_1<1}\prod_{i=1}^kx_i^{m-\frac{k+1}{2}}\prod_{i<j}(x_i-x_j)\exp(-\sum_{i=1}^kw_ix_i)}{\int_{0<x_k&...
4
votes
0answers
80 views
Korovkin subset of $C(\mathbb{T})$
Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
6
votes
1answer
251 views
Almost complex manifold of dimension 2… locally isomorphic to ℂ?
I know that this is supposed to be standard, but I don't know how to search for it... hence the question:
Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\...
2
votes
1answer
106 views
Conformal mappings and its singularity
I have a question about singularities of conformal mappings.
Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to ...
5
votes
0answers
42 views
First Cousin Problem for Bergman spaces
I recall (an easy case of) the first Cousin problem :
Let $\Omega_1, \Omega_2$ be two open subsets of the complex plane
$\mathbb{C}$ with non-empty intersection and $f$ be holomorphic on $\...
2
votes
0answers
47 views
Criteria for a limit to be a proper function
This question is obviously broad; turning this broadness into something sharp is part of the problem.
Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
2
votes
0answers
47 views
Positivity and zeros of Heun's function
I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. Consider the case where the Frobenius indices at $0$ are $(0,1- \...
3
votes
0answers
17 views
Complex Matrix Model with an External Field
I have the following random matrix model
$$ Z = \int[dU dU^\dagger] e^{-N\text{Tr}S}, \quad S = UU^\dagger + gX(UU^\dagger + U^\dagger U) + \frac{g}{3}(U^3 + (U^\dagger)^3) $$
where $X$ is a ...
4
votes
0answers
224 views
Is every algebraic curve the critical set of an algebraic function?
Is every algebraic curve in $\mathbb{R}^2$($\mathbb{C}^2$), the set of critical points of a polynomial in $\mathbb{R}[x,y]$($\mathbb{C}[x,y]$)?
In particular what is a real (complex) polynomial whose ...
1
vote
0answers
67 views
Is the normalized derivative of a holomorphic function Sobolev?
This is a cross-post.
Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic on the interior $\text{int}(B)$, and smooth on the closed disk $B$.
...
2
votes
1answer
93 views
Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$
If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$
An approximate solution of $\phi$ ...
4
votes
2answers
272 views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
0
votes
0answers
27 views
Reference Request: Carnot Groups over Complexes
Is there a theory of complex (analytic) Carnot groups and Caratheodory metrics?
2
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0answers
83 views
Integral of complex rational functions in several variables
Let $D\subseteq \mathbb{C}^3$ be an open domain, and let $f:D\to \mathbb{C}$ be a rational function. That is: $f$ can be written as a quotient $f=g/h$ where $g,h\in \mathbb{C}[z_1,z_2,z_3]$.
Question:...
4
votes
1answer
197 views
Normal Cones for Complex Spaces
Suppose $U\subset\mathbb C^n$ is an open subset and $f_1,\ldots,f_k$ are analytic functions on it, generating the coherent ideal sheaf $\mathcal I$ which defines a closed complex subspace $Z\...
4
votes
2answers
191 views
Multiplication in Deligne cohomology: explicit formula for $p=q=1$
[This is a double of my question of math.stackexchange https://math.stackexchange.com/questions/3214962/multiplication-in-deligne-cohomology-explicit-formula-for-p-q-1]
In the very beginning of [1] ...
2
votes
1answer
85 views
Ratio of hypergeometric function
Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function?
$$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$
for $1>x>y>0$ ideally in the form like some ...
3
votes
1answer
158 views
Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
16
votes
1answer
570 views
Dirichlet series with a single zero
I need to find a Dirichlet series f that has the following property.
f is zero in only one point s such that Re(s) > $\sigma_c $.
14
votes
1answer
1k views
Floor of Riemann zeta function
How to show that $$\left\lfloor\zeta\left(1+\frac{1}{n}\right)\right\rfloor=n$$ for every positive integer $n$?
2
votes
0answers
77 views
automorphic form associated with Apollonian Gasket
In /Indra's Pearls/, it's mentioned one can associate automorphic forms with limit sets. Is there an explicit description of the one associated with the Apollonian gasket (up to some appropriate ...
3
votes
0answers
106 views
Constructing a model of $\mathrm{DCF}_0$ via forcing
As is mentioned in the introduction of this paper of Spodzieja there is a lack of 'natural' examples of differentially closed fields. The immediate naive guesses, namely the field of germs of ...
4
votes
0answers
66 views
Algebraization of holomorphic functions of two variables
Let $f: \mathbb C ^2\to \mathbb C$ be (a germ at $0$ of) a holomorphic function. Does there exist a small neighborhood $U_0\in \mathbb{C}^2$ of $0$ and a holomorphic change of coordinates $g$ (i.e. ...
6
votes
1answer
159 views
complex manifold with boundary
My question is of local nature.
Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.
Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)...
1
vote
0answers
48 views
The analyticity on a closed ball
In complex analysis, how to understand statements like a fuction $f$ is analytic on $\bar B(0,1)$ ? I cannot figure out the notion of analyticity at a boundary point. Thank you.
0
votes
0answers
124 views
Generating function with essential singularities
I was recently introduced to analytic combinatorics, and found the method of removing poles astonishing. More precisely, I was reading the last chapter of the popular "generatingfunctionology", in ...
0
votes
0answers
61 views
Question on h-infinity norm of a system
Consider a control system,
$\dot{x}=Ax+Bu\\
y=Cx$.
Define the transfer function $G(s)=C(sI-A)^{-1}B$.
Then it is claimed that the following definitions of induced norm are equivalent.
$\|G\|_{\...
6
votes
1answer
320 views
Tight error terms for partial sums $\sum_{n\leq x} 1/n^s$
(a) Let $s>1$, $x>0$ be real. Then it is not hard to see that
$$\sum_{n\leq x} \frac{1}{n^s} \leq \zeta(s) - \frac{1}{(s-1) x^{s-1}} + \frac{1}{2 x^s},$$
basically because $x\mapsto 1/x^s$ is ...
5
votes
1answer
223 views
Find a function $F$ on $[0,1]$ with moments decaying as $(\ln n)^{-n}$
Let $F:[0,1]\to\mathbb{R}$ be a measurable function such that
$$
\mu_n(F)=\int_0^1F(t)t^ndt\sim\frac1{(\ln n)^n}\quad\mbox{as}\quad n\to\infty.
$$
More precisely,
$$
0<c<|\mu_n(F)|(\ln n)^n<...
2
votes
1answer
52 views
Euclidean length of hyperbolic geodesics for annuli with bounded geometry
I am wondering whether there are estimates for the Euclidean length of vertical hyperbolic geodesics for annuli with good geometry.
More precisely:
Take an annulus $A$, whose outer boundary $\gamma_{...
2
votes
1answer
154 views
families of Riemann mappings
Let $U\subset \mathbb R^n$ be an open.
Let $f: S^1 \times U \to \mathbb C$ be a smooth map (i.e. $C^\infty$) s.t. for every $x\in U$ the map $f_x:=f|_{S^1\times \{x\}}:S^1\to \mathbb C$ is a smooth ...
1
vote
0answers
62 views
Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
1
vote
0answers
72 views
An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
5
votes
1answer
147 views
Notational question about quadratic differentials in Strebel's book “Quadratic differentials”
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
2
votes
1answer
56 views
Intersection of superlevel set of polynomials
Let $P_1$ and $P_2$ be complex polynomials with complex coefficients and $c > 0$. Can we find polynomial $P_3$ and $c’>0$ such that
$\{z \in \mathbb C : |P_1(z)| \geq c\} \cap \{ z \in \...
10
votes
1answer
468 views
Is there a holomorphic function on open unit disc with this property?
Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ ...
6
votes
0answers
109 views
How to interpret Gauss's late fragments on conformal mapping of the interior of an ellipse (to the unit disk) in modern mathematical terms?
My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the ...
2
votes
0answers
70 views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
1
vote
0answers
50 views
Identities for beta functions and twisted cohomology
This is a question about notation, I apologize if it is too basic. In the paper
Cho, Koji; Matsumoto, Keiji, Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I, ...
3
votes
1answer
151 views
An integral inequality for diffeomorphisms
Assume that $F(e^{it})=e^{if(t)}$ is a diffeomorphism of the unit circle onto itself and let $A=\left|\int_0^{2\pi}(1-F^2)\,dt\right|$ and $B=\left|\int_0^{2\pi} F^2(1-F^2) \,dt\right|$. It seems that ...
3
votes
1answer
118 views
Is the disk algebra a complemented subspace of the algebra of bounded analytic functions?
It is well known that the disk algebra (viewed as an algebra on the circle) is uncomplemented in $C(\mathbb T)$. What can be said about the pair
$(A(\mathbb D), H^\infty(\mathbb D))$?
0
votes
0answers
36 views
integrating multivariable rational function over a product of disks
Suppose I have a rational function of $k$ complex variables:
$$ R(x_1,...,x_k) = P(x_1,...,x_k)/Q(x_1,...,x_k) $$
where $P$ and $Q$ are polynomials. Now I'd like to compute the integral of this ...
11
votes
0answers
155 views
Holomorphically convex manifolds and Bergman complete manifolds
Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is ...
1
vote
1answer
174 views
A question about the proof of Riesz-Thorin interpolation theorem
I was reading the proof of Riesz-Thorin interpolation theorem in http://www.math.kit.edu/iana3/lehre/fourierana2014w/media/rieszthorinproof.pdf
and get stuck at the last step. We construct the complex ...
2
votes
0answers
46 views
Calculus over Function Fields of Characteristic Zero
Having done some cursory searching of the internet, it is clear to me that there is a very well-developed theory of how to do calculus over function fields, such as fields of Laurent series in a ...
3
votes
1answer
196 views
Why are Poincare series defined as they are?
We know the Poincare series are defined as the following:
The $m^{th}$ Poincare series of weight $k$ for $\Gamma$ is:
$$
P_{m}^{k} (z) = \sum_{(c,d)=1} (cz+d)^{-k} e^{2 \pi in(\tau z)}.
$$
The ...
1
vote
0answers
131 views
Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$
Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$:
$$
V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}.
$$
...
4
votes
2answers
225 views
Sums of entire surjective functions
Suppose $(f_n)_n$ is a countable family of entire, surjective functions, each $f_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a_n)_n$, not all zero, such that $\sum_{n=1}^{\...
3
votes
3answers
183 views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
6
votes
1answer
178 views
Do analytic functionals form a cosheaf?
Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
