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

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2
votes
0answers
47 views

What are “minimal lines” in complex geometry?

Schwarz reflection across a real analytic arc $C$ in $\mathbb{R}^2$ is usually defined analytically. Thinking of the arc as the image of an interval in $\mathbb{R}$ under an invertible holomorphic map ...
3
votes
2answers
171 views

Conformal map and Jordan curve

Here is my question : Suppose you have a simple (analytic) closed curve $\gamma$ in an open simply connected domain $\Omega \neq \mathbb{C}$. Does there exist a conformal bijection $f : \Omega ...
7
votes
0answers
181 views
+100

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
4
votes
2answers
263 views

Non-bijective conformal maps between annuli

I need to answer the following question, hopefully in the negative. Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk ...
1
vote
0answers
44 views

Related to derivative of Modified Bessel I function wrt the order

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$. Using maple, it seems that ...
0
votes
1answer
39 views

A Generalization of growth exponents

The growth exponent of a function $f(\sigma + it)$ is defined as the least nonnegative real number $\psi(\sigma)$ satisfying $$ f(\sigma + it) \ll |t|^{\psi(\sigma) + \epsilon} $$ as $t \to \infty$, ...
-3
votes
0answers
119 views

So what exactly is the deal with Lewy's example and Misohata's Example?

I was considering Lewy's example in the theory of Partial Differential Equations over the complex plane: http://en.wikipedia.org/wiki/Lewy%27s_example Which states that for certain functions ...
-4
votes
0answers
37 views

A mapping from the group PSL(2,R) [closed]

...that sends $z_1$ to i and $z_2$ to some place on the vertical line $iR^+$, given any two arbitrary points from the upper half complex plane. Any ideas? Thanks.
1
vote
1answer
45 views

Oscillation of subharmonic functions of slow growth

Given a sequence of real numbers $c_k\to-\infty$, is there always a $C^\infty$ subharmonic function $f$ on $\mathbb R^2$ and a sequence $z_k\to\infty$ with $|z_k|<k$ such that ...
3
votes
2answers
285 views

Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements. Let $h,k\geq1$ be integer numbers and let ...
0
votes
0answers
40 views

Hurwitz's theorem for a system of functions

First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$. We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow ...
86
votes
42answers
29k views

Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
5
votes
0answers
52 views

Extremal length of graphs in surfaces

Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be $$ \sup_{g \in \omega} ...
0
votes
0answers
54 views

Derivative of a conjugation of matrices

Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...
3
votes
1answer
459 views

Is the integral always nonzero?

Let $$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$ where $$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < ...
1
vote
1answer
166 views

Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...
1
vote
1answer
100 views

Interesting property of analytic functions

Let $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$, be an analytic application, such that: $f(t)=0\Longleftrightarrow\ t=t_0$. Is it true that there is an analytic function $g:(t_0-\varepsilon, ...
0
votes
0answers
61 views

prove that a function is approximatively three dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$. Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...
1
vote
0answers
45 views

Analytic continuation of an integral

Let $$f(y)=\frac{y_1^{1/3}y_2^{1/3}}{y_1+y_2+1}$$. Consider the following integral: $$F(s_1,s_2)=\int_{\mathbb{R}_+^2}f(y)^{s_1}f(y^{-1})^{s_2}\frac{dy_1dy_2}{y_1y_2}$$ where ...
7
votes
2answers
163 views

Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
9
votes
1answer
266 views

Higher Fano varieties and Tsen's theorem

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...
5
votes
1answer
312 views

Bounds on the derivative of a Riemann map

Let U be a simply connected domain with smooth boundary in the complex plane, and let $\mathbb D$ be the unit disc. Is there a nice sufficient condition for the existence of a biholomorphic map ...
3
votes
2answers
250 views

Residue for the generating function of the Euler totient function

Let $\varphi$ be the Euler totient function, and let us define the function $f(z)$ by the series $$ f(z) := \sum_{n=1}^{\infty} \varphi(n) z^n $$ Since $0\le \varphi(n)\le n$, I believe this gives a ...
0
votes
0answers
36 views

Euler transformation of pochhammer symbol

From the Euler transformation of Pochhammer symbol $$\sum_{n=0}^{\infty}\frac{(b)_n}{n!}a_nz^n=(1-z)^{-b}\sum_{n=0}^{\infty}\frac{(b)_n}{n!}\Delta^na_0(\frac{z}{1-z})^n$$ the following ...
1
vote
0answers
46 views

Geodesic equation and radial metric

Assume that $g(z)=f(|z|)$ is a radial metric on the unit disk in complex plane, where $f$ is a smooth real function. Is there any simple equation of geodesic lines w.r.t. metric $g$, e.g. ...
30
votes
1answer
625 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
0
votes
3answers
127 views

Extension of conformal map and annulus

My question is the following : suppose you have a doubly-connected open set $\Omega \subset \mathbb{C}$, that is a domain bounded by 2 non-intersecting circles $C_1$ (the interior) and $C_2$ (the ...
2
votes
1answer
244 views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
16
votes
2answers
419 views

Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
1
vote
1answer
92 views

If $f$ is separately holomorphic on $\Omega$ then $f\in\mathcal{C}^0(\bar\Omega)\Leftrightarrow f\in L^1(\Omega)$

Let $\Omega\subseteq\Bbb C^2$ be open bounded (and connected), $f:\Omega\to\Bbb C$ separately holomorphic (i.e. $f$ is holomorphic in each variable when the other is fixed). Hartogs theorem is not ...
6
votes
1answer
265 views

Acyclicity of the sheaf of real analytic differential forms

Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that the Poincare lemma holds for the de Rham ...
1
vote
2answers
80 views

Integrability at $z$ of the 2-form $ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $

Given $g\in\mathcal{C}^1(\bar\Delta)$, and $z\in\Delta$, how can i prove that the 2-form $$ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $$ is integrable in $z$? At ...
7
votes
0answers
149 views

When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?

If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when is ...
27
votes
1answer
1k views

Computing Self-Intersections with Complex Analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis: $$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of $f(z) = 0$ inside a close ...
3
votes
1answer
131 views

intersection of holomorphic curve with hyperplane

Let $f : \mathbb{C} \rightarrow \mathbb{C}^n$, $n>1$ be an entire function. Assume for simplicity that $f(0)=0$. Let $B$ be the closed ball of centre $O$ and radius $R$. Is there an upper bound ...
1
vote
1answer
114 views

Zeros of Polynomial with decreasing coefficients [closed]

If $n_1<n_2<n_3\cdots<n_m$ are positive integers. Does the polynomial $a_0+a_1z^{n_1}+a_2z^{n_2}+\cdots+a_mz^{n_m}$ satisfying $$ 0<a_0\leq a_1\leq \cdots\leq a_m $$ has all its zeros ...
3
votes
1answer
77 views

About extending plurisubharmonic function

I came across a problem like this. Suppose that $\Omega$ is an open subset of $\mathbb{C}^{n}$ and $V$ is a complex submanifold of $\Omega$ of codimension 1. Now given a plurisubharmonic function ...
7
votes
3answers
1k views

When is a holomorphic submersion with isomorphic fibers locally trivial?

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...
1
vote
2answers
127 views

convergence radius of Pochhammer symbol series

What is the convergence radius of the series $$\sum_{n=0}^{\infty}\frac{(-1)^n z^{(n)}}{n!},$$ where $z^{(n)}$ is Pochhammer symbol?
1
vote
1answer
125 views

generalization of fundamental theorem of algebra for several complex algebra [closed]

I am looking for a generalization to fundamental theorem of algebra for several complex variables functions or systems. If such theorem exists, it should concisely relates the number of zeros of ...
1
vote
0answers
36 views

Meromorphic extensions of $\zeta-$functions

Suppose we have a series $\zeta_x(s)=\sum_{n\geq1}x(n)n^{-s},$ where $x=\{x(n)\}_{n\geq1}$ is a bounded sequence. Clearly, $\zeta_x$ is analytic function for $\Re(s)>1.$ Question: is there a ...
2
votes
2answers
169 views

Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result: Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$ be a non ...
0
votes
0answers
24 views

Is the Mellin transform of a measure nongrowing at imaginary infinity everywhere, or just on the fundamental strip?

Let $\mu$ be a measure on the positive real numbers. Its Mellin transform is a complex function defined by $$ M_\mu (s) =\int x^{s-1} d \mu(x) $$ on the set $S_\mu$ of $s \in \mathbb{C}$ where $$ ...
8
votes
1answer
175 views

Continuous Weierstrass map

Let $\mathbb C$ be the complex plane, $H(\mathbb C)$ the set of all entire functions, and $D(\mathbb C)$ the set of all non-negative divisors in $\mathbb C$. Consider the map $Z:H(\mathbb C)\to ...
14
votes
2answers
293 views

Vanishing of Dolbeault cohomologies and Steinness

That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
1
vote
2answers
141 views

Generalized Schwarz Lemma for near-zeros

In approximation theory, it is classical to use a result that can be considered a generalization of the Schwarz Lemma: Let $f:[-1,1]\rightarrow\mathbb{C}$ be a function that is analytic in a domain ...
13
votes
2answers
346 views

Analog of Newlander–Nirenberg theorem for real analytic manifolds

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost ...
0
votes
1answer
91 views

Dirichlet series without order term

is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions? $D(s) = \sum_{0<n<N}a_n/n^s$ The ...
1
vote
0answers
55 views

Complex Hessian Signature

It' all, simply, about the signature of a matrix. Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
46
votes
16answers
10k views

f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential

The question is about the function f(x) so that f(f(x))=exp (x)-1. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here ...