Tagged Questions

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why is this result about Gaussian analytic functions equivalent to the Crofton formula

I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function. Definition A Gaussian analytic function ...
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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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Mittag-Leffler function and Laplace Integral

Let $E_{\alpha}(z)\triangleq \sum_{n=0}^{\infty} \frac{z^n}{\Gamma(\alpha n + 1)}$ be the Mittag-Leffler function. I am looking for a full proof of the following fact (a reference to a proof in the ...
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Magnitude of the sum of complex i.u.d. random variables in the unit circle

Hello everybody. I'm working about asymptotic estimates of $M_n = \left|\sum_{k=1}^n Z_k\right|$ where $Z_1, Z_2, \ldots$ are independent uniformly distributed random variables on the complex unit ...
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Traceless GUE : Four Centered Fermions

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel \[ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
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What is the orthonormal basis for the Bergman space on the disk?

[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.] In arXiv:0310.5297, ...
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Polynomials on the Unit Circle

I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...
Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...