Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,152
questions
2
votes
1
answer
838
views
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
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z \right|...
2
votes
1
answer
404
views
Monotonicity and perturbation of $J$-holomorphic curves
In a symplectic manifold $(M, \omega)$, $J$-holomorphic curves are special minimal surfaces if $J$ is compatible with the symplectic structure and the metric is induced by $\omega$ and $J$.
Minimal ...
2
votes
2
answers
322
views
Pointwise bounds on Hardy space functions with regular boundary behaviour
Let $H^2$ denote the Hardy space on the strip $S:=\{z\in{\mathbb C}\,:\,0<\Im z <1\}$ (or the upper half plane), i.e. $H^2$ consists of all holomorphic functions $f:S\to\mathbb C$ such that for ...
2
votes
4
answers
310
views
polynomial zero within a square
Is there any complex polynomial $p$ of one variable having no zeros within the unit square $-1 < \Re(z) , \Im(z) < 1$ such that $\left|p(0)\right|$ is strictly smaller than $\left|p(z)\right|$ ...
2
votes
1
answer
150
views
Entire functions of exponential type with small $L^1$ norm outside a finite real interval
I'm interested in entire functions of exponential type $\sigma$ (Bernstein space $B_\sigma^1$) following
$$\int_{-\infty}^{\infty} |f(x)|dx=1,$$
whose norm is as small as possible outside a range $[-...
2
votes
2
answers
352
views
On the set of zero radial limits of bounded analytic functions
Hi,
Let $f$ be a non-identically zero bounded analytic function in the open unit disk $\mathbb{D}$. It is well-known that $f$ has radial limits almost everywhere on the unit circle $\mathbb{T}$. Let $...
2
votes
4
answers
1k
views
Nth root of a matrix as an analytic function?
Let $A$ be a $k \times k$ invertible matrix over complex numbers.
If it possible to write its nth root as an analytic function (i.e. power series in $A$)?
EDIT: Complex coefficients can be functions ...
2
votes
2
answers
312
views
Factorization of antisymmetric bounded holomorphic functions
A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = ...
2
votes
1
answer
96
views
Are the real components of s-roots subharmonic?
Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative ...
2
votes
1
answer
877
views
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 ...
2
votes
1
answer
137
views
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+...
2
votes
1
answer
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 ...
2
votes
1
answer
981
views
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 ...
2
votes
1
answer
678
views
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\...
2
votes
1
answer
193
views
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|...
2
votes
2
answers
383
views
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 ...
2
votes
1
answer
102
views
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\...
2
votes
1
answer
330
views
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*}
...
2
votes
3
answers
434
views
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 ...
2
votes
1
answer
86
views
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.$$
Call the corresponding real reproducing kernel ...
2
votes
1
answer
161
views
Existence of the special entire Hardy space function with infinitely many zeros in the strip
Question. Does there exist an entire function $h$ satisfying three following assertions:
$h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
$zh - 1$ belongs to $H^2(\mathbb{C}...
2
votes
1
answer
190
views
Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
2
votes
1
answer
90
views
Singularity on the boundary of domain of holomorphy
Let $\phi$ be a continuous function on the closed upper half-plane $\{ z\in\mathbb{C}: \operatorname{Im}(z)\ge 0\}$ and holomorphic in the interior.
Suppose that the function $x\phi(x)$ is in $C^1(\...
2
votes
1
answer
122
views
Regarding basis of holomorphic Hardy space
Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain and let $H^2(\partial\Omega)$ denotes a holomorphic Hardy space which is a $L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)(=\...
2
votes
1
answer
548
views
Mellin transform of powers of gamma function
If $a>0$, the Cahen-Mellin integral gives
$$\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}^{\...
2
votes
1
answer
208
views
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} \...
2
votes
1
answer
213
views
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 $\mathbb C$.
...
2
votes
1
answer
229
views
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 ...
2
votes
1
answer
159
views
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 ...
2
votes
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{...
2
votes
1
answer
175
views
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 $$\...
2
votes
1
answer
120
views
Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$
Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$.
Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of
$$p(w)=n+\sum_{j=1}^{m}\frac{...
2
votes
1
answer
174
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 $...
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
$$
\mu_E(A)=\frac{1}{\inf\{\|X\|:X\in E\text{ and }\det(I_n-...
2
votes
1
answer
1k
views
Bromwich integral transformed to an integral on the real axis
I am new in complex integration and inverse Laplace transforms. I already asked this question on math.se but got no answer.
The author of a textbook claims that the inverse Laplace transform has ...
2
votes
1
answer
342
views
Riemann mapping theorem, boundary
The Riemann mapping theorem states that for every non-empty, simply connected, open set $\Omega \subset \mathbb{C}$, which is not all of $\mathbb{C}$ there exists a function $f$ that maps $\Omega$ ...
2
votes
1
answer
185
views
Regarding outer function being the quotient of two outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
2
votes
1
answer
109
views
Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$
If I have the following integral equation $$\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 $\phi$ ...
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....
2
votes
1
answer
349
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 ...
2
votes
1
answer
89
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 ...
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.
2
votes
1
answer
152
views
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 ...
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 ...
2
votes
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 \...
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 ...
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 $\mathbf{U}$, ...
2
votes
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 ...
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 ...
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 ...