# Tagged Questions

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

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### 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 ...
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### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by ...
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### 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 $0<\beta<1$.
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### 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 ...
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### Is the Euler product formula always divergent for 0<Re(s)<1?

It is known that the Euler product formula converges for Re(s)>1. (which represents the Riemann zeta function.) My question: Is the Euler product formula always divergent for 0 < Re(s) < 1 ? ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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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### 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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### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function). Is there any conceptual explanation - or ...
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### Can Voronin's universality theorem be used to show that $\sigma\circ\zeta=\zeta\circ\sigma$ implies $\sigma$ continuous?

Let $\sigma$ be a field automorphism of $\mathbb{C}$ that commutes with the Riemann Zeta function. Can we use Voronin's universality theorem to prove that $\sigma$ is necessarily continuous? Thanks in ...
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### 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 ...
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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). ...
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### 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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### 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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### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties:  f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...