# Tagged Questions

**1**

vote

**1**answer

165 views

### System of quadratic complex equations

I want to solve this system of N non-linear equations without using a numerical method:
$x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$
With
$\left| ...

**0**

votes

**0**answers

202 views

### Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help.
The integral itself is, with $\tau$, $\lambda$, and ...

**4**

votes

**0**answers

135 views

### 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 ...

**2**

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**0**answers

198 views

### Analytical continuation of electrostatic potentials

I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the electrostatic ...

**1**

vote

**1**answer

232 views

### 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 ...

**0**

votes

**1**answer

361 views

### The real and imaginary parts of the Incomplete Gamma function for second argument being purely imaginary

Dear all,
I am looking for explicit (at least more explicit than the original expression) for
1) Re$(\Gamma(a, i\omega))$
as well as
2) Im$(\Gamma(a, i\omega)),$
where i Re and Im denote the real ...

**3**

votes

**1**answer

140 views

### Analytic continuation of instantaneous eigenstates of a time-dependent hamiltonian

We are considering the instantaneous eigenstates of an analytically time-dependent hamiltonian and I would like to know how legitimate it is to extend them to the complex plane.
Specifically, our ...

**4**

votes

**3**answers

362 views

### What classes of functions are closed under all rescalings?

Let us denote by the symbol $\mathcal{G}$, a group of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ (with the composition operation) that is additionally closed under all affine change of variables ...

**3**

votes

**0**answers

264 views

### About a Christoffel-Darboux-type sum

Hi!
I've been using the Christoffel-Darboux identity for the Hermite polynomials,
$$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^n n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$
for some ...

**2**

votes

**0**answers

317 views

### Gaussian type integral with inverse square root

Hi,
I have encountered an integral of the following type in an engineering application:
$\int_{-\infty}^\infty dx \frac{1}{\sqrt{x^2+a^2}}\exp(-x^2/2+i x b)$,
where $a$ and $b$ are real ($a$ could ...

**3**

votes

**1**answer

119 views

### quadrature domains from circles?

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 ...

**5**

votes

**3**answers

422 views

### 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!} ...

**1**

vote

**1**answer

214 views

### S-transformation of generalized Eisenstein series

I'm currently using generalized Eisenstein series to construct weight 2 modular forms under $\Gamma_1(N)$. They are defined as
$E_2^{\psi,\phi}(\tau) = \delta(\psi) L(-1,\phi) + 2\sum_{n=1}^{\infty} ...

**4**

votes

**1**answer

390 views

### Vandermonde-type identity for Jacobi theta functions?

My question concerns an application in physics. By Vandermonde identity I refer to the following statement: take $f_j (z)=z^j$, where $z=x+iy$ is a complex coordinate and $j$ an integer. Make an ...

**8**

votes

**0**answers

365 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**9**

votes

**1**answer

608 views

### Approximation to divergent integral

Hi everyone,
I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), ...