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2
votes
0answers
250 views

Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?

Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to ...
7
votes
1answer
518 views

Non-algebraic curve visualisation

Is there any software which can automatically visualise a non-algebraic complex curve, I mean the structure of it's ramification points and sheet? I think a good test example would be the Lambert ...
1
vote
1answer
353 views

Holomorphically Convex Hull a Subset of the convex hull of

This comes from Hörmander's "An Introduction to Complex Analysis in Several Variables". We defined the $A(\Omega)$-hull (analytic functions in an open set $\Omega$). $\hat{K}$ of a compact set ...
1
vote
1answer
270 views

Necessary condition for a branch point

If I have a function $f(z,\alpha)$ (let's keep it a polynomial of order $\geq 2$ in $z$, for simplicity), what would be a necessary condition for there to be branch points for this function? A friend ...
11
votes
3answers
1k views

What holomorphic functions are limits of polynomials?

Let $\Omega$ be a connected open set in the complex plane. What is the closure of the polynomials in $\mathcal{H}(\Omega)$ the set of holomorphic functions on $\Omega$? The topology is the usual ...
1
vote
2answers
329 views

Minimum number of polygonal lines connecting points in an annulus.

It is obvious that an open annulus in the complex plane: $S = a < |z| < b$ is connected. That is, each pair of point $z_1$ and $z_2$ in it can be joined by a polygonal line. What is the minimum ...
2
votes
1answer
394 views

Homology of a region of the plane

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let $$ ...
9
votes
0answers
1k views

Meaning of Cauchy integral theorem - the (co)homology viewpoint

I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try. In the elementary theory of analytic functions of $1$ complex variable, one ...
6
votes
3answers
1k views

Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ? Edit: To rule out the case ...
0
votes
1answer
380 views

Proving uniform bound

Hello I want to prove that $\lim_{h\rightarrow\infty}\left(\int_{0}^{\infty}\left(\cos ...
9
votes
1answer
831 views

When are complex polynomial maps almost surjective?

Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$. For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
1
vote
1answer
256 views

how to prove the relationship between pseudoconvexity and the monge-ampere matrix?

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if ...
6
votes
1answer
234 views

Orbits for homogenous complex polynomials under unitary rotation of variables

Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$. ...
1
vote
1answer
578 views

Irreducible non-singular M-matrices and complex numbers

It is well known that a non-singular M-matrix that is irreducible has a strictly positive inverse (all entries $>0$). An M-matrix is a matrix that has eigenvalues with positive real part, and the ...
0
votes
1answer
372 views

Topology and convergence for conformal maps of disk

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, centre $0$. Write $\mathcal{S}$ for the holomorphic injective maps $\{ f : \mathbb{D} \to \mathbb{D} | f(z) = e^{-\lambda} z + O(z^2) \}$ i.e. ...
4
votes
1answer
427 views

Polylogarithm inequality

Recall the polylogarithm $Li_s(z)=\sum_{n=1}^\infty \frac{z^n}{n^s}.$ For $|z|<1,$ is $\Re \left( \frac{Li_1(z)}{Li_2(z)} - \frac{2}{3}\frac{Li_2(z)}{Li_3(z)} \right)>0?$ The numerics suggest ...
3
votes
2answers
874 views

Proper holomorphic map from unit disk to punctured unit disk

It is easy to see that there cannot be a proper holomorphic map from the punctured unit disk to the unit disk in the complex plane. What about the other direction; does there exist a proper ...
4
votes
3answers
584 views

Is there a “Riemann mapping theorem” for a circle in C^2 ?

The Riemann mapping theorem says that if you have a simple closed curve in $\mathbb{C}$, then there is an essentially unique way to map a holomorphic disc to the interior. Is there any reasonable ...
7
votes
1answer
1k views

Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following: 1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
4
votes
3answers
565 views

magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal

What can you say about a function defined on a square region of the complex plane, if the integral of the function along any horizontal, vertical or diagonal of the square is equal ? - an analytic ...
2
votes
0answers
231 views

Question in complex analysis arising from large $N$ gauge theory

This is a question in complex analysis that comes up in the treatment of large $N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...
2
votes
0answers
595 views

Pole data of meromorphic matrix function

Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable. Recall that such a $T$ is said to have a right pole of order $r$ ...
2
votes
1answer
1k views

Relation between complex analysis and harmonic function theory [closed]

There are some theorems in harmonic function theory that resemble results in complex analysis, like: Holomorphic functions and complex functions are analytic; Cauchy's integral formula in complex ...
10
votes
2answers
1k views

Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently: In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations ...
1
vote
1answer
278 views

Conjugate Groups of (quasi) Fuchsian Groups

I apologize in advance if this question is so trivial or too low level. Let $\Gamma$ be a Fuchsian group. Let $\mathcal{F}$ be the set of pairs $(\mu,f)$, where $\mu \in L^\infty(\mathbb{C})$ such ...
9
votes
1answer
477 views

How manifold-like is Aut(C^n) in the holomorphic category?

This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just ...
2
votes
2answers
617 views

What $Re(f(z))=c$ can be if $f$ is a holomorphic function ?

Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function. Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then ...
1
vote
0answers
115 views

Relation between different spatial derivatives of a random field (related to complex integral and/or bessel function)

2 random fields $b$ and $c$ are derived from random field $a$ by $b=\nabla^2a\equiv(\partial_{xx}+\partial_{yy})a $ and $c \equiv c_1+i c_2 = (\partial_{xx}-\partial_{yy}+2i \partial_{xy}) a$. ...
1
vote
3answers
759 views

A simple ordinary differential equation

Consider an entire function $f : \mathbb{C} \rightarrow \mathbb{C}$! We search the function $$ g: (a,b) \rightarrow \mathbb{C},$$ which solves the following equation locally: $g'(t)=f(g(t))$ and ...
7
votes
3answers
882 views

Conformal Mappings for hyperbolic polygon

I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics. The classical Schwarz Christoffel theorem does the job ...
8
votes
3answers
2k views

Harmonic level sets and boundary data

This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great: Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary ...
54
votes
1answer
3k views

Is there a “classical” proof of this $j$-value congruence?

Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve ...
2
votes
1answer
207 views

Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?

Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a ...
2
votes
1answer
265 views

Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex?

Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex? Also, if X is a Stein space with a finite ...
3
votes
3answers
566 views

Analytic continuation via square of absolute value

Is the following fact true (I think that I can prove it but I don't trust myself on these matters): let $f(z)$ be an analytic function defined on some open subset $U$ of ${\mathbb C}$. Assume that the ...
5
votes
1answer
3k views

Meaning of \Subset notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
10
votes
5answers
2k views

Complex Analysis applications toward Number Theory

I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory. So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...
3
votes
1answer
417 views

methods for interpolating a function, holomorphic in the upper halfplane

Let $n,k\colon\mathbb{R}\to\mathbb{R}$ be real functions such that function $N$ given by $N(x)=n(x)-ik(x)$ is a holomorphic function in the upper half-plane. Also I know some additional properties of ...
7
votes
1answer
656 views

Solving the Beltrami Equation for a very simple Beltrami Coefficient

Let $\mu$ be a function on the complex plane with the property $\mu(z) = \overline{\mu(\bar{z})}$, such that $\mu(z) = \epsilon e^{-2\pi i \bar{z}}$ on the upper-half plane, where $\epsilon$ is a ...
3
votes
1answer
392 views

Factorization in the Wiener algebra on the unit disc.

Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where $$f(z)=\sum ...
13
votes
2answers
866 views

Complete metric on the space of Jordan curves?

I was interested in putting a complete metric on the space of Jordan curves. Say, just planar Jordan curves contained in $B(\bar{0}, 2) \backslash B(\bar{0}, 1)$ which separates $\bar{0}$ and ...
2
votes
0answers
217 views

Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer. Background: When ...
18
votes
4answers
1k views

What is the naming reason of poles in complex analysis?

A function $f: \textbf{C} \to \textbf{C}$ has a pole of order $k$ if $f(z) = \frac{g(z)}{(z-z_0)^{k}}$ where $g(z)$ is a nonzero analytic function. Why do we call it poles?
2
votes
1answer
847 views

The normal derivative of the Green's function

I was wondering if anything was known about the following: Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk. Consider now the Green's functions $G(z; p)$ ...
1
vote
2answers
489 views

Weierstrass Theorem [closed]

Hi-- Where can I find a proof of this theorem: For each $r \in \mathbb{Z}_{+}$, there exists a complex entire function $f(z)$ such that $f(r) \neq 0$ but $f(r+1)=f(r+2)=\cdots =0$, i.e. $f(z) \in ...
14
votes
5answers
960 views

What is the spectrum of the ring of entire functions?

Let $\mathcal{O}(\mathbb{C})$ be the ring of entire functions, that is, those functions $f : \mathbb{C} \to \mathbb{C}$ which are holomorphic for all $z \in \mathbb{C}.$ For each $z_0 \in \mathbb{C}$. ...
4
votes
0answers
322 views

Adeles of Holomorphic Functions

In number theory, an adele is a kind of product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...
6
votes
2answers
2k views

Functions of several complex variables: book recommendations?

Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...
7
votes
1answer
709 views

Dirichlet series expansion of an analytic function

Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$ ...
47
votes
32answers
9k views

Demystifying complex numbers

At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are ...