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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Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is $$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z \right|...
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Monotonicity and perturbation of $J$-holomorphic curves

In a symplectic manifold $(M, \omega)$, $J$-holomorphic curves are special minimal surfaces if $J$ is compatible with the symplectic structure and the metric is induced by $\omega$ and $J$. Minimal ...
Guangbo Xu's user avatar
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Pointwise bounds on Hardy space functions with regular boundary behaviour

Let $H^2$ denote the Hardy space on the strip $S:=\{z\in{\mathbb C}\,:\,0<\Im z <1\}$ (or the upper half plane), i.e. $H^2$ consists of all holomorphic functions $f:S\to\mathbb C$ such that for ...
Gandalf Lechner's user avatar
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4 answers
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polynomial zero within a square

Is there any complex polynomial $p$ of one variable having no zeros within the unit square $-1 < \Re(z) , \Im(z) < 1$ such that $\left|p(0)\right|$ is strictly smaller than $\left|p(z)\right|$ ...
5th decile's user avatar
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Entire functions of exponential type with small $L^1$ norm outside a finite real interval

I'm interested in entire functions of exponential type $\sigma$ (Bernstein space $B_\sigma^1$) following $$\int_{-\infty}^{\infty} |f(x)|dx=1,$$ whose norm is as small as possible outside a range $[-...
Jesus's user avatar
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2 answers
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On the set of zero radial limits of bounded analytic functions

Hi, Let $f$ be a non-identically zero bounded analytic function in the open unit disk $\mathbb{D}$. It is well-known that $f$ has radial limits almost everywhere on the unit circle $\mathbb{T}$. Let $...
Malik Younsi's user avatar
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Nth root of a matrix as an analytic function?

Let $A$ be a $k \times k$ invertible matrix over complex numbers. If it possible to write its nth root as an analytic function (i.e. power series in $A$)? EDIT: Complex coefficients can be functions ...
Piotr Migdal's user avatar
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Factorization of antisymmetric bounded holomorphic functions

A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = ...
Tobias Hartnick's user avatar
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Are the real components of s-roots subharmonic?

Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative ...
Daniel Parry's user avatar
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The version of Montel's theorem used in the proof of Jenkins-Strebel differential

Hello, I am afraid that my main question might be a bit too elementary, but still I ask : In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...
Analysis Now's user avatar
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Discriminant on boundary of semi-algebraic surface

Let $P(t)$ be a polynomial in $t$ of degree $n$, with some contiguous coefficients (not the first or last) being $x_1,\dots,x_k$ and the rest of the coefficients are fixed. (E.g. $p(t)=1+2t+x_1t^2+...
Per Alexandersson's user avatar
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324 views

A Fact Of Quasiconformal Map

We just consider puntured unit disk $\triangle^{*}$ in $\mathbb{C}$. $f$ is a bounded quasiconformal map on $\triangle^{*}$. Why $f$ can extend to the origin,becoming quasiconformal map on the whole ...
Quanting Zhao's user avatar
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A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane

I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or ...
Analysis Now's user avatar
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A basic question on the definition of Cartan-Remmert reduction and holomorphic convexity

Here is a definition of holomorphic convexity taken from the notes of Eyssidieux: Defintion. A complex analytic space $S$ is holomorphically convex if there is a proper holomorphic morphism $\pi: S\...
aglearner's user avatar
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Coefficients of lacunary series on quasiconformally transformed unit disk

Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if $|q|...
graveolensa's user avatar
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Question concerning minimum of hyperbolic metric

Hello all, I am interested in the following question. Suppose a,b,c,z are points in the complex sphere. Consider the family of curves g through a,b,c, and for each g let U be the complement of g in ...
Greg Markowsky's user avatar
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Functions with asymmetrically decreasing Fourier transform?

$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ with $u,v:\bbR\to\...
TaQ's user avatar
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Combinatorial meaning of a binomial expansion

Let $F$ be a generating function $F(x) = \sum_{i=0}^\infty f_i x^i$, and suppose that we can do operations formally without worrying about convergence issues. Define the coefficients \begin{gather*} ...
Student's user avatar
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Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?

Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function? How about the positivity, monotonicity, and convexity of the ...
qifeng618's user avatar
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Pair of positive harmonic functions with negative inner product in Drury-Arveson space

Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel ...
J. E. Pascoe's user avatar
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Existence of the special entire Hardy space function with infinitely many zeros in the strip

Question. Does there exist an entire function $h$ satisfying three following assertions: $h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane; $zh - 1$ belongs to $H^2(\mathbb{C}...
Pavel Gubkin's user avatar
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190 views

Linear elliptic equation

Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
Samir's user avatar
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Singularity on the boundary of domain of holomorphy

Let $\phi$ be a continuous function on the closed upper half-plane $\{ z\in\mathbb{C}: \operatorname{Im}(z)\ge 0\}$ and holomorphic in the interior. Suppose that the function $x\phi(x)$ is in $C^1(\...
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Regarding basis of holomorphic Hardy space

Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain and let $H^2(\partial\Omega)$ denotes a holomorphic Hardy space which is a $L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)(=\...
Naruto's user avatar
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548 views

Mellin transform of powers of gamma function

If $a>0$, the Cahen-Mellin integral gives $$\DeclareMathOperator{\Res}{\operatorname{Res}} \frac{1}{2\pi i}\int\limits_{a-i\infty}^{a+i \infty}\varGamma(z) u^{-z}dz=e^{-u}=\sum\limits_{m=0}^{\...
Mark's user avatar
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A 2 dimensional integral in polar coordinate [closed]

Recently I got stuck on a 2 dimensional integral in polar coordinate, the expression is the following: $I(x)=\lim_{\xi\rightarrow0^+}\int_0^\infty dr\int_{-\pi/2}^{\pi/2}dt\frac{2\xi ^{2-2 x}r^{2x+1} \...
NuKuYul's user avatar
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How to characterize the images of disk-algebra functions?

It is well known that the continuous images $f:\mathbf D\to \mathbb C$ of the closed unit disk $\mathbf D$ are exactly the non-void compact, connected, locally path connected sets in $\mathbb C$. ...
ray's user avatar
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Bicomplex Conjugate Derivative

I have decided to first ask my question and second provide a list of steps I have already considered. Question: After reading Luna-Elizarrarás, Shapiro, Struppa, and Vajiac - Bicomplex numbers and ...
Talmsmen's user avatar
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Max-root inequality for convex combination of real-stable monic polynomials (Kadison-Singer Problem)

In the paper "The Kadison-Singer Problem" by Marcin Bownik (https://arxiv.org/pdf/1702.04578.pdf), the following Lemma (3.8) is proven: Lemma: Let $p, q\in \mathbb{R}[x]$ be stable monic ...
Strickland's user avatar
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1 answer
117 views

Twisted winding number

Consider the contour integral $\frac{1}{2\pi i}\oint_\gamma\chi(z)\frac{dz}{z}\,,$ where $\gamma$ is a (not necessarily simple) closed curve lying in $\mathbb{C}\setminus{0}$ and $\chi\colon\mathbb{...
Jack L.'s user avatar
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Existence of a distinguished continuous version of the logarithm of a continuous function

Let $E$ be a $\mathbb R$-Banach space and $\varphi\in C^0(E,\mathbb C\setminus\{0\})$ with $\varphi(0)=1$. I want to show that there is an unique $\psi\in C^0(E,\mathbb C)$ with $\psi(0)=0$ and $$\...
0xbadf00d's user avatar
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1 answer
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Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$

Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$. Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of $$p(w)=n+\sum_{j=1}^{m}\frac{...
Thomas Kojar's user avatar
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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 $...
asrxiiviii's user avatar
2 votes
1 answer
149 views

Regarding upper semicontinuity of a function

Let $E$ be a linear subspace of $\mathbb{C}^{n\times n}$. Define the function $\mu_E:\mathbb{C}^{n\times n}\longrightarrow \mathbb{R}_+$ as $$ \mu_E(A)=\frac{1}{\inf\{\|X\|:X\in E\text{ and }\det(I_n-...
user429197's user avatar
2 votes
1 answer
1k views

Bromwich integral transformed to an integral on the real axis

I am new in complex integration and inverse Laplace transforms. I already asked this question on math.se but got no answer. The author of a textbook claims that the inverse Laplace transform has ...
Stéphane Laurent's user avatar
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1 answer
342 views

Riemann mapping theorem, boundary

The Riemann mapping theorem states that for every non-empty, simply connected, open set $\Omega \subset \mathbb{C}$, which is not all of $\mathbb{C}$ there exists a function $f$ that maps $\Omega$ ...
H. Rittich's user avatar
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1 answer
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Regarding outer function being the quotient of two outer functions

Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
user31459's user avatar
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1 answer
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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 $\phi$ ...
Mahmoud Hassan's user avatar
2 votes
1 answer
338 views

Defining integrals by residue theorem

I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....
Penchez's user avatar
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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 ...
JJH's user avatar
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1 answer
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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 ...
user131261's user avatar
2 votes
1 answer
184 views

Roots of unity and an extremal problem [closed]

I want to determine the subset of $m$ members ($m < n/2$) of the set $e^{i 2\pi k/n}, \ \ k=0,\dots, n-1$, so that the absolute value of its sum is maximal.
Lira's user avatar
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conformal mapping and rational function

Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. By Riemann mapping theorem, there exists a unique exterior ...
Masik Kara's user avatar
2 votes
1 answer
620 views

The Borel-Laplace transform of a transeries that contains logarithms

I am interested in Ecalle's generalization of the Borel-Laplace summation. I would like to see an explicit treatment of a summation of a transeries that include logarithmic terms. The only example I ...
tst's user avatar
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1 answer
342 views

Mean value theorem in terms of Wirtinger calculus?

The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as \begin{equation} f(x)-f(y)=\int_{0}^{1}\nabla f(x(\tau))d \tau \...
Wuchen's user avatar
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2 votes
1 answer
120 views

Attempts to Solve the Euclidean TSP in the Complex Plane

Question: have there been any serious (meaning by a reputated mathematician) attempts to solve the euclidean TSP in the complex plane by interpreting the $(x,y)$ coordinates of the real plane as ...
Manfred Weis's user avatar
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2 votes
1 answer
218 views

Absolutely continuous and rectifiable boundary

Assume that $\gamma$ is a Rectifiable curve in $\mathbf{C}$ and ssume that $f$ is a bounded holomorphic function on the unit disk $U$ such that if $z_n$ converges to a boundary point of $\mathbf{U}$, ...
KDist's user avatar
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1 answer
180 views

Modulus of Continuity for an Analytic Function on an Ellipse

Given $f\in C^{\infty} (E)$, where $E\subseteq \mathbb{C}$, define $E_{\rho} \subseteq \mathbb{C}$ as the maximal ellipse with foci at $\{-1,1\}$ where $f$ is analytic, and semi-minor + semi-major ...
Amir Sagiv's user avatar
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2 votes
1 answer
193 views

Control of a meromorphic function according to distance between its zeros

My question is rather philosophical : can a meromorphic function of normalized norm with simple zeros on the flat torus stay close to zero on a large set when its zeros are far from each other ? The ...
Fougeroc's user avatar
2 votes
1 answer
143 views

Higher dimensional analogue of Ahlfors covering surface theory

It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher ...
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