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

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3
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65 views

Stokes's Theorem with singularities on projective line

Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity: $$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
4
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1answer
325 views

An open problem on polynomials

Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$ It is an open problem and I did not find any ...
2
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41 views

Holomorphic semigroups vs analytic semigroups

Is there any difference between the two notions in the theory of semigroups? In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...
2
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57 views

Derivative of a polynomial $P(z)$ and the derivative of the conjugate reciprocal of $P(z)$

Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|<1.$ Let $Q(z)=z^n\overline{P(1/\overline{z})}.$ Then it is an easy exercise to show that $\Re\left(zP'(z)/P(z)\right)= \sum_{k=1}^n\...
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72 views

Theta-function in the lower half-plane

Standard theta function $$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$ has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
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27 views

Fixing coefficients using analytic structure

I am trying to understand an exercise in a set of lecture notes on random matrices - Eynard - Random matrices - given in the paragraph following (4.6.60) (pp. 70–71). Specifically, I am trying to fix ...
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1answer
60 views

Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
2
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1answer
76 views

Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
3
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1answer
75 views

A question on preimage of a locally injective meromorphic function

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If ...
3
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1answer
185 views

To show holomorphicity of a certain infinite series of functions

I have the following sequence of holomorphic functions on $(f_n(s))_{n \geq 1}$ on the closed region $R:= \{ s \in \mathbb C : \Re(s) \geq 1\}$ by $$f_n(s) := \begin{cases} \log(1+p^{-s})-p^{-s}\...
2
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205 views

On the logarithmic derivative of an analytic function

I have reached upto this situation while exploring some properties of analytic functions which are bounded on the unit circle. Suppose $f(z)$ be such an analytic function in the closed unit disc ...
4
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1answer
199 views

Fun examples relating to Hopf surfaces

A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...
3
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1answer
110 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
10
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1answer
303 views

Sendov's conjecture

It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...
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51 views

zero extension of positive currents are always positive

In Demailly's Complex Analytic and Differential Geometry page 139: He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
6
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1answer
105 views

The state of art of the singular Levi problem — and hyperkähler quotients

One of the versions of the classical Levi problem asks the following: Let $X$ be a complex manifold. Is it true that $X$ is Stein iff $X$ admits a smooth exhaustion strictly plurisubharmonic ...
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49 views

Resources for divergent / asymptotic series

This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside Other than the usual references given in Wikipedia and Mathworld, which resources have you found ...
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39 views

maximum modulus of polynomials

Suppose $p(z)$ is a polynomial of degree $n$ having no zeros in $|z|<1.$ Then for any $R\geq1$ we have a result $ \max_{|z|=R}|p(z)|\leq \frac{1+R^n}{2}max_{|z|=1}|p(z)|,\;\;R\geq 1.$ I could not ...
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48 views

Separating a Riemann-Hilbert problem

Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is $$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$ where $g_j(x)$ are Holder continuous functions and $...
3
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84 views

Exponential iterates of a complex number

Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$. In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...
3
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1answer
81 views

Space of holomorphic functions multiplied by smooth functions taking real values

Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...
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1answer
50 views

weak convergence of positive currents vs. $L^1$ convergence of normalized potentials

I have run into the following statement in the literature (e.g. here, p.5, after Theorem 1.1): that weak convergence of positive $(1,1)$-currents on a complex manifold is equivalent to $L^1$ (I ...
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43 views

Locally connectedness and accessibility in $\mathbb{C}$

Suppose $\Omega$ to be a bounded area in the complex plane $\mathbb{C}$ with a locally connected boundary $\partial\Omega$, then every point of $\partial\Omega$ is accessible from Ω.Here accessibility ...
8
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1answer
333 views

Complex plane minus Cantor set admits non-constant bounded harmonic function

Let $K\subset [0,1]$ denote the usual 1/3 Cantor set. I know that $\mathbb{C}\backslash K$ has no non-constant bounded analytic function, since the singularity $K$ can be removed. However, a statement ...
3
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63 views

Construction of weight function to satisfy condition on given functional

(Sorry for similar and trivial looking question ; But could have potential application in prime number theory ) Consider the following function : $$F(z) = \omega(z)\frac{\sin^2\left(\frac{c\Gamma^2(...
2
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2answers
352 views

if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?

This question is related to solving $f(f(x))=g(x)$. Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
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1answer
184 views

Entire even functions of order 1 have infinitely many zeros?

Let $f$ be an entire even function of order 1 such that $f(0)\neq 0$. Does $f$ have infinitely many zeros?
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147 views

cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be. For example, say $T^4$ is regarded as a complex ...
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0answers
150 views

What is decoupling theory means on Tao Blog ? And what is its purpose in mathematics? [closed]

I accrossed on Tao Blog a new theory for me which it is called "Decoupling theory", But I didn't find in the web its definition and its purpose , I find only this article in wiki but this very far ...
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59 views

How to solve a problem from Frank Olver's book

I'm learning Frank Olver's book, called Asymptotics and Special Functions. There is an difficult exercise. Problem. Suppose that $f,\frac{1}{f}$ possess the following asymptotic expansions : $$f(...
6
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3answers
438 views

Acting with all rational rotations on a subset of the circle having positive measure do you fill almost the whole circle?

Set $\Gamma$ for the group of the roots of the identity: $\Gamma=\{z\in \Bbb C | z^n=1$, for some $n\geq 0\}$ and for $E\subset S^1$ set $\Gamma E=\{z\zeta, z\in \Gamma, \zeta\in E \}$ A trivial but ...
3
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0answers
76 views

Existence of a maximal rank CR Lie subalgebra

Let $\mathfrak{g}$ be a real Lie algebra of dimension $2n+1$ and let $\mathfrak h \subset \mathfrak g \otimes \mathbb C$ be a subalgebra of complex dimension $n+1$ satisfying $\mathfrak h + \overline{\...
4
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1answer
67 views

Rate of convergence of Padé approximants

Let $f$ be an entire function of order $1$. Two questions: 1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)? 2) if yes, can ...
1
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1answer
41 views

Stieltjes transform of a compactly supported measure : behaviour at the boundary

I ask here the same question I asked on Mathematics to maybe reach other poeple : I am studying the Stieltjes transform $$ G_\mu(z) = \int_a^b \frac{1}{z-s} d \mu(s) $$ of some positive finite ...
4
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0answers
117 views

What is the closed form of this integral?

Consider the Chebyshev first function $\psi(y):=\sum_{p^j \leq y} \log p$, where $p$ is a prime. Define $$F(s, k) = s\int_{1}^{\infty} \psi(x + x^k)x^{-s-1} \mathrm{d}x$$ for $ \Re(s) >$ max $(1, ...
2
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1answer
77 views

A density question

Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that $$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = ...
2
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2answers
93 views

Reference request for the integral representation of the Hadamard product of two infinite series

Define $F(x) = \sum_{n\geq 1} f_{n}x^n$ and $G(x) = \sum_{n\geq 1} g_{n}x^n$. Then the Hadamard product of $F$ and $G$ is $$H(x):=(F*G)(x) = \sum_{n\geq 1} f_{n}g_{n}x^n.$$ The author of Riesz ...
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0answers
36 views

Asymptotics of a certain integral in singularity theory

Let $f:\mathbb{C}^2\to \mathbb{C}$ be an isolated plane curve singularity. Consider the versal deformation space $\mathbb{C}^\mu$ parameterizing deformations $f_\lambda$ for $\lambda \in \mathbb C^\mu$...
2
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0answers
72 views

Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ ...
3
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1answer
64 views

Is a domain biholomorphic to the unit ball a Runge domain?

Let $\Omega \subset \mathbb C^n$ be a bounded domain which is biholomorphic to the unit ball $B^n=\{|z|<1 \mid z\in \mathbb C^n\}$. Can we show $\Omega$ must be a Runge domain? By definition, $\...
2
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1answer
112 views

Subharmonic in any holomorphic coordinates = Plurisubharmonic?

An upper semi-continuous function $u : \Omega \to \mathbb{R}$, $\Omega \subseteq \mathbb{C}^n$ is said to be subharmonic if it satisfies the submean inequality $u(a) \leq \mu_S(u;a,r)$, where $\mu_S(...
3
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0answers
50 views

Limit of Hankel function for large complex order, fixed real argument

Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...
2
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1answer
75 views

Conformal isomorphism uniquely determined by boundary identification?

Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\...
3
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1answer
102 views

Reference on boundary behavior of conformal maps

I am looking for some results on the boundary behavior of conformal maps between simply connected domains. In particular I am interested in conformal maps between $\mathbb{C}-\Delta$, where $\Delta$ ...
0
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1answer
89 views

How to check if you have the asymptotic solution of some equation? [closed]

Suppose I have an analytic function $f : \mathbb{R} \to \mathbb{R}$ and I have the asymptotic expansion of some $x_0$ up to a few terms in terms of $\epsilon$ for some $\epsilon \to 0$ which I believe ...
0
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1answer
73 views

Transformation which “opens up” an arc

I am reading Harold Widom's paper "Extremal Polynomials Associated with a System of Curves in the Complex Plane". At the beginning of section 11 he states that: [There is] a simple transformation ...
3
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1answer
131 views

Isoperimetric inequality for analytic functions on an annulus

Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that $$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
8
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1answer
518 views

A question on the Riemann zeta function

Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer,...
1
vote
1answer
119 views

Commuting matrices of complex functions

If $A(z) :=[A_{ij}(z)] $ and $B(z) :=[B_{ij}(z)] $ are two invertible $n\times n$ matrices of entire complex valued functions entries $A_{ij}(z)$, and $B_{ij}(z) $ with (1). $AA^{\#}=A^{\#}A$ ...
10
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1answer
225 views

Convex Julia sets

Consider the classical Julia set $J_f$ associated with $f(z)=z^2+c$. Since $J_c$ is completely invariant, we know that $f^{-1}(J_f) \subseteq J_f$. Now, let $H_f$ be the convex hull of $J_f$. Is it ...

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