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How does one write down $\mathbb{R}$-valued functions on the modular surface? I am considering taking an arbitrary function on the upper half plane $f:\mathbb{H} \to \mathbb{R}$ and averaging over th …

I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant: $$ \Delta(\theta) = \prod_{i< j} |e^{i\theta …

If $h(z)$ is analytic on the disk centered at 0 of radius r, by the Cauchy Residue formula \[ \int \int_D h(z)\, dx dy = \pi r^2 h(0) \] The disk is the simplest example of a quadrature domain since …

I am reading notes on a complex interpolation problem: Let $z_1, \dots, z_n \in \mathbb{D}$ and $w_1, \dots, w_n \in \mathbb{C}$. There exists (bounded holomorphic?) $f \in H^\infty(\mathbb{D})$ w …

In a probability theory paper I found this rather pleasant result: Theorem 4.1 Let $n \geq 2$ and $f_1, \dots, f_n : \mathbb{C} \to \mathbb{C}$ be meromorphic with possible poles at $\{ \mathfrak{ …

We learn in complex analysis class how to find the winding number of the contour $\Gamma$ around the origin. \[ n = \frac{1}{2\pi i} \oint \frac{dz}{z} = \frac{1}{2\pi i} \oint d(\log z) = \fra …

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 …

Basically, I am trying to build explicit examples of Dirac operators. To this end, I'm looking at the surface E = C/(Z + λZ) - for some λ in H \ SL(2,Z) - with the Euclidean metric and the flat conne …

The Julia set of the map $z \mapsto z^2+i$ is a dendrite fractal. I would like to know which affine maps (other than identity) map this region to a subset of itself. I imagine there are two three ge …

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis: $$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of $f(z) = 0$ inside a close cur …

[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, Yu …

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 …

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!} …