# Tagged Questions

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

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### On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$ $L^{1}/H^{1}_{0}$ is ...
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### Are there entire functions that are unexpectedly periodic? [on hold]

I wonder if there have ever been entire functions discovered/defined that turned out to be periodic and the periodicity was unexpected and surprising to mathematicians ?
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### No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$

See David Speyer's answer here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, ...
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### 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 $M$ ...
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### What is the name of this theta related function? [on hold]

My adviser showed me a function which is important for my current work, but he doesn't remember where he got this. He just knows it's a kind of theta function. I cannot find any literature talking ...
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### 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 ?
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### 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 ...
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### 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 ...
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### 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 ...
155 views

### Argument that seems to indicate the Riemann Hypothesis could be false? [closed]

I haven't got any deep understanding of the zeta function or anything that complex. I just noticed the following computation. We know $\frac{1}{\zeta(z)}=\sum_{n\geq 1}\frac{\mu(n)}{n^z}$. Now let ...
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### Interchange of summation and analytic continuation with absolute convergence and an analytic sum? [migrated]

Let $f_n$, $F$ be analytic on the open right half plane, and suppose that for all real $x > 0$ and all integers $k \ge 0$ the series $\sum_{n=1}^\infty f_n^{(k)} (x)$ converges absolutely to ...
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### 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 ...
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### conformal mapping of exterior region and Osgood-Taylor-Caratheodory theorem [migrated]

Osgood-Taylor-Caratheodory theorem deals with extending continuously a conformal mapping between interior domains of Jordan curves in $\mathbb{C}$ to the boundary(see here for an original proof). ...
634 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 ...
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### 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 ...
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### 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 ...
232 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 $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
492 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 ...
180 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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, ...