# Tagged Questions

**2**

votes

**0**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**42**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**16**answers

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 ...