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4
votes
1answer
419 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
828 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
568 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
549 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
224 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
582 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 ...
9
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
269 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
473 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
610 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
752 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
865 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 ...
52
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
203 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
262 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
533 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 ...
9
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
618 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
386 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
836 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
214 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
828 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
482 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
921 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
318 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
667 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 ...
1
vote
2answers
431 views

Extension of harmonic function at infinity

Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property ...
5
votes
0answers
536 views

some questions about properties of harmonic measure

The original post The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ...
2
votes
2answers
474 views

L^2 space of holomorphic functions with given weight

Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product $\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar ...
8
votes
1answer
678 views

When are entire functions surjective?

Is there some useful criterion to determine whether or not an entire function is surjective?
3
votes
3answers
551 views

Analytic ODE with complex time

Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r. I would like to understand: 1) if there exists an analytic flow ...
-4
votes
1answer
445 views

Meaning of the Mobius transformations video [closed]

What is this video trying to tell us? http://www.youtube.com/watch?v=JX3VmDgiFnY The statement that fractional linear transformations correspond to rotations of the sphere under the stereographic ...
11
votes
2answers
998 views

Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

The following describes an extension of the Descartes Rule of Signs to polynomials with complex coefficients. First, I need to define the notion of a "sweep"... Given a complex polynomial p(z) := c0 ...
5
votes
2answers
299 views

Can curves induced by analytic maps wiggle infinitely across a line?

Let $f$ be a function analytic on an open subset $D\subset \mathbb{C}$, and let $\gamma:[0,1] \to D$ be a line segment. $g = f\circ\gamma$ is another curve in the complex plane; is it possible to for ...
2
votes
0answers
284 views

What is this effect in Fourier/additive synthesis called?

Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were ...
2
votes
2answers
1k views

Branched coverings of Riemann surfaces with specified branch points.

Today I showed, using some ad hoc algebraic topology, that if $\Sigma$ is a Riemann surface and $\mathfrak{p} \subset \Sigma$ is a finite set of points, then there is another Riemann surface $S$ and a ...
5
votes
1answer
2k views

How would You encourage graduate students to learn algebraic geometry and/or complex analysis?

Hello, I am the 3rd year undegraduate student of mathematics. After I obtain a bachelor degree I want to study maths at graduate level, especially algebraic geometry and complex analysis. This fields ...
3
votes
2answers
917 views

Upper half plane quotient by a discrete group

I was reading Mehta and Seshadri's paper "Moduli of vector bundles on curves with parabolic structures". In the second paragraph, they wrote: "Suppose that $H$ mod $\Gamma$ has finite measure ($H$ ...
1
vote
0answers
644 views

Bessel function in polar coordinates

I want to write the Bessel function of the first kind in polar coordinates $J_\alpha(z)=|J_\alpha(z)|e^{i\varphi_\alpha(z)}$ Is anything known about $\varphi_\alpha(z)$? In particular, I'm ...
3
votes
1answer
380 views

Finding Functional form for a given Scaling Condition

Dear all While studying the overlap distribution for two random Cantor sets (long story made short), I came across the following problem. $G(k)$ is a complex valued function, and satisfy the ...
2
votes
1answer
224 views

Help determining the asymptotic behavior of an integral involving rational functions.

Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that $$\int_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu ...