# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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### 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&...

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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\|_\...

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**1**answer

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 $\...

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**1**answer

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 ...

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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 $\...

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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 ...

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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- \...

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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 ...

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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 ...

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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 $B$.
...

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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 $\phi$ ...

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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{\...

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27 views

### Reference Request: Carnot Groups over Complexes

Is there a theory of complex (analytic) Carnot groups and Caratheodory metrics?

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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:...

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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\...

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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] ...

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**1**answer

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 ...

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**1**answer

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_{...

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**1**answer

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 $.

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**1**answer

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 $n$?

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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 ...

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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 ...

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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. ...

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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)...

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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.

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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 ...

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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\|_{\...

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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 ...

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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<...

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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_{...

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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 ...

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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 ...

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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 ...

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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 ...

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**1**answer

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 \...

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**1**answer

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 $f$ ...

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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 ...

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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 ...

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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, ...

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**1**answer

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 ...

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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
$(A(\mathbb D), H^\infty(\mathbb D))$?

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ${\mathbb C}^n$:
$$
V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}.
$$
...

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votes

**2**answers

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}^{\...

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**3**answers

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 ...

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**1**answer

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 ...