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

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27
votes
3answers
816 views

A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set $$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$ What can be said about the set $M$ ...
4
votes
0answers
113 views

classification of homogenous complex manifolds

Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?
2
votes
0answers
56 views

Hadamard Product of specific type of power series

I am consider the power series of the form $$F_n(t):=\frac{1}{\prod_{i=1}^n(1-t^i)}.$$ Given two power serires $A(t)=\sum_{i\ge 0}{a_it^i}$ and $B(t)=\sum_{i\ge0}{b_it^i}$, their Hadamard product is ...
13
votes
3answers
435 views

Completeness of nonharmonic Fourier Series

I have the following question: The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$. Thus, certainly the oversampled system $\Phi:...
0
votes
2answers
139 views

Explicit analytic function with modulus asymptotic to $\Re z+\Im z$

Is there a simple and explicit continuous function $f\colon[0,\infty)^2\to\mathbb C$ such that $f$ is analytic on $(0,\infty)^2$ and $|f(x+iy)|/(x+y)\to1$ as $x+y\to\infty$, where $(x,y)\in[0,\infty)^...
11
votes
1answer
442 views

$\pi e$ and an unfamiliar polynomial

Ever since my exposure to this integral involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral $$\int_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, \...
1
vote
0answers
56 views

Inverse Laplace Transform involving irrational powers

Could anybody please suggest a reference or a possible solution how to invert the Laplace transform of $e^{-(s^{\alpha}+\lambda)^{\beta}}$, where $0<\alpha<1$ and $0<\beta<1$.
0
votes
1answer
150 views

Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra

Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$ that it inherits as the dual of $L^{1}(\...
11
votes
1answer
193 views

Is there a proof of the uniformization theorem using circle packing?

In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same ...
2
votes
1answer
152 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
2answers
141 views

Asymptotics of the derivatives of analytic functions

Are there sources that treat questions like the following ones? Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as $x\...
2
votes
2answers
301 views

H. Cartan's “Variétés analytiques complexes et cohomologie”?

Does anyone know where I might find an online version (for free or purchase, translated or in french) of this paper by Henri Cartan from 1953? I know it was published in Colloque sur les fonctions de ...
2
votes
0answers
39 views

A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261). I've tried posting it on MSE, and placed a generous bounty, but I couldn't get any answers there. *3. Using Ex. 2, show that $...
6
votes
1answer
137 views

Factorization of conformal maps between annuli

Consider two doubly-connected open subsets $A$ and $A'$ of the Riemann sphere. We assume these two domains to be of same modulus (the moduli space being one real parameter), i.e. we assume that there ...
8
votes
2answers
262 views

A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$

Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a ...
18
votes
1answer
302 views

Positivity of coefficients of the inverse of a certain power series

Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation $$ g(z)-g(z)^8+g(z)^{15}=z, $$ that is the inverse of $$ z-z^8+z^{15} $$ in the group of formal ...
5
votes
1answer
149 views

On a Sum of Gamma Functions

I am working on a problem where the following sum appears: $$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum_{n=0}^{\infty}{\frac{s^{n} t^{n}}{\left[(s+1)(t+1)\right]^{n+1+\alpha}}\frac{\Gamma(n+1+2\alpha)}{\...
8
votes
1answer
332 views

Removing singularities in generating functions

This is a problem about the practicalities of removing singularities in multivariable complex functions. In trying to derive the generating function (in two variables) for a certain problem in ...
4
votes
0answers
79 views

Concluding that the Poisson kernel is indeed the Cauchy distribution?

See here. Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
2
votes
0answers
140 views

Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$ [closed]

I have been playing around with Mathematica and continued fractions and I noticed something. ContinuedFractionK[n, n + x, {n, 1, Infinity}] ==-x + 1/(E Gamma[1 + x] - E Gamma[1 + x, 1])==-x + 1/(E ...
2
votes
0answers
77 views

Poisson kernel, follow-up question, follows that process $\left\{e^{i\theta X_t - \theta Y_t}\right\}$ is a martingale? [closed]

See here. Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. For any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{i\...
7
votes
4answers
671 views

Is an entire function, with nowhere vanishing derivative, always a covering map?

Assume that $f:\mathbb C\to\mathbb C$ is entire, and also that $f'(z)\ne 0$, for all $z\in\mathbb C$. Does that imply that $f$ is a covering map of $f[\mathbb C]$? Clearly, $f$ is a local ...
7
votes
2answers
316 views

What are some important papers that use complex analytic techniques to get good bounds?

The motivation behind this question is somewhat similar to that of the tricky project launched by Gowers et al, but is certainly a specialization. My work tends to rely on both exact formulae and ...
7
votes
2answers
196 views

How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root

(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...
5
votes
1answer
238 views

harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
2
votes
0answers
180 views

Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
1
vote
1answer
176 views

Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane

Setting: Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$. Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
3
votes
0answers
130 views

polynomial relations between modular functions

$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$ We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
1
vote
0answers
69 views

Construction of homogeneous Siegel domain from j-algebra

I am reading bounded homogeneous domain from Piatetski-Shapiro's book ``Automorphic functions and the geometry of classical domains'' and have questions on how to construct homogeneous Siegel domain ...
2
votes
4answers
164 views

A question on Ahlfors covering surface

Given a transcendental entire function $f$, and three Jordan domains $D_1$, $D_2$, and $D_3$ such that the closures of the three Jordan domains do not intersect with each other. Then from Ahlfors ...
5
votes
0answers
88 views

Finite covers in complex analytic geometry

Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ...
16
votes
1answer
669 views

Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let $\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals. Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{...
0
votes
1answer
83 views

Hilbert transform on boundary value of analytic bounded functions

I am considering the boundary values of a bounded holomorphic functions. Suppose $w$ is a bounded holomorphic function in upper half plane, with continuous and bounded boundary value $f$ on real axis. ...
10
votes
1answer
226 views

Tori in three-space

Recently I was talking to an alien who does not know complex function theory. I was trying to convince her that the set of conformal equivalence classes of smooth embedded tori in $R^3$ is two ...
11
votes
0answers
261 views

Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty f_p(x)x^{s-1}dx=e^{-p\psi(s)} $$ where $\psi(s)$ is the usual digamma function. The inverse Mellin formula ...
1
vote
0answers
85 views

To show there exists a unique function $u \in C^{1}(\mathbb{C^n})$ that satisfies $(\bar{\partial u})=f$

Assume $n \gt 1$. Let $f$ be a $(0,1)$ form in $\mathbb{C^n}$, with $C^1$-coefficients and compact support $K$, such that $\bar{\partial} f=0$. Let $\Omega_{0}$ be the unbounded component of $\mathbb{...
23
votes
2answers
590 views

Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction. Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...
8
votes
1answer
211 views

Angular distribution of zero sets of sparse polynomials

Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily ...
8
votes
0answers
323 views

rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index. Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index. Let $M_0(\Gamma_i)$ denote the ring of modular ...
3
votes
1answer
74 views

compact almost complex submanifolds of complex Lie groups

I find the following Corollary 1.21: Question: does there exist any complex Lie groups $G$ such that there are some compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$? I want ...
0
votes
0answers
148 views

Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments. Maybe you guys can help. http://math.stackexchange.com/questions/1440931/...
12
votes
0answers
307 views

Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
1
vote
0answers
56 views

Zeros of functions constituting a Riesz-basis for the Paley-Wiener space

I have a short question which first requires some slightly elaborate definitions: Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...
13
votes
3answers
470 views

How bad can a circle domain get?

Let $X$ be a domain in the Riemann sphere $\widehat{\mathbb{C}}$. We say that $X$ is a circle domain if every connected component of its boundary is either a circle or a point. It was conjectured by ...
3
votes
1answer
274 views

Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)

Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$) $$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x t^{...
2
votes
0answers
188 views

Hartogs's extension theorem

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$, and let $f:H\to \mathbb{C}^n$ be a holomorphic injective map. Then we know that $f$ extends holomorphically to the polydisc $P$, i.e. there ...
2
votes
0answers
121 views

Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their ...
14
votes
2answers
831 views

Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex? When does the topological ...
1
vote
2answers
93 views

classification of rational map with exactly only one Fatou component

We know that there exists a polynomial the Fatou set $F(P)$ is connected, which is just an attracting basin for infinity. I have a question: Given a rational function $R$ such that $F(R)$ is ...
2
votes
2answers
83 views

Is there a non-Shih analog for holomorphic functions of the Intermediate Value Theorem?

Let $C$ be a simple closed curve in the complex plane, and let $f$ be holomorphic on an open set containing $C$. Is there a condition on the signs of Im $f$ and Re $f$ on $C$ that guarantees the ...