# Tagged Questions

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### M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...
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### On the geometry of roots of a sum of complex linear fractions

I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form $$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$ where the $a_j$'s are nonzero complex ...
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### semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix. \[ \int ...
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### Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form \begin{equation*} f(t)=\sum_{k=-n}^n c_k ...
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### A counterexample to the Polya-Schur master theorems for half-planes

Given an integer $n\ge 1$ we say that $f\in C[z_1,\ldots,z_n]$ is stable if $f(z_1,\ldots,z_n)\neq 0$ whenever $\text{Im}\ z_i>0$ for all $1\leq i\leq n$. Stable polynomials with all real ...
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### Entire functions of exponential type with small $L^1$ norm outside a finite real interval

I'm interested in entire functions of exponential type $\sigma$ (Bernstein space $B_\sigma^1$) following $$\int_{-\infty}^{\infty} |f(x)|dx=1,$$ whose norm is as small as possible outside a range ...
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### Asymptotic behavior of entire functions

Which entire function $f\left(x\right)$ goes asymptotically to $\dfrac{e^{-x}}{x}$ as $x$ goes to infinity with $x$ positive? That is, $\left(e^{-x}/x \right)/f \left(x \right) \rightarrow 1$.
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### Effective vanishing of the Schwarzian Derivative

Recall for any complex analytic function $f:\mathbb{D}\to \mathbb{C}$ the Schwarzian derivative of $f$ is $$S(f)=\frac{f'''}{f'}-\frac{3}{2} \left( \frac{f''}{f'}\right)^2.$$ It's well known that ...
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### Complex evaluation of a classical (real) integral

There are several ways to compute the classical integral $$\int_{\mathbb R}e^{-x^2}dx=\sqrt{\pi}.$$ Probably, best known are (1) squaring the integral with subsequent change of (now two) variables ...
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### Show that holomorphic functions are infinitely differentiable without complex analysis [duplicate]

Possible Duplicate: Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions? Is there a way to show that holomorphic ...
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### Analytic implicit function theorem

I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic ...
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### holomorphic equation

hi, i am working for some time on a problem and at some point i cant go further. here the critical part: Let $U \subset \mathbb{C}^{n}$ be a open set and consider $c : U \rightarrow \mathbb{R}$ a ...
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### Class of flat currents stable under $\overline{\partial}$ operator

Given $U\subset\mathbb{C}^n$, open domain, a locally flat current on $U$ is a $k-$current $T$ such that for every $f\in\mathcal{D}(U)$ (smooth functions with compact support in $U$) there exist a ...
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### Boundary behavior of a holomorphic function on $D$ ?

Hi, I have two related questions. $D$ = open init disk in the complex plane $C$. A. Let $f: D \to C$ be a holomorphic function. Then is it possible that $\forall q \in S^1$,there exists a ...
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### Approximation by analytic functions

Dear all. Let $$f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)$$ be a function given by usual fourier series. Since my original question hasn't got any answer yet, and I ...
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### Asymptotic approximation of $x^\alpha$ by entire functions

Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$ for $x\rightarrow+\infty$ (with ...
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### Current status of Bloch Constant and Landau Constant bounds

The Bloch constant B (based on a theorem introduced by AndrĂ© Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance ...
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### Most important domains, extension theorems, and functions in several complex variables

For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) ...
I want to know when certain expressions of the form ${\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)}$ are algebraic numbers. These ...