相关论文: Asymptotics of Landau constants with optimal error…

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and…

Let $\mu_1$ and $\mu_2$ be two complex-valued Borel measures on the real line such that $\operatorname{supp} \mu_1 =[\alpha_1,\beta_1] < \operatorname{supp} \mu_2 =[\alpha_2,\beta_2]$ and ${\rm d}\mu_i(x) = -\rho_i(x){\rm d}x/2\pi {\rm i}$,…

In this note, we study weak solutions of equation \begin{equation}\label{eq 00.1} \Delta u =\frac{4e^u}{1+e^u} -4\pi\sum^{N}_{i=1}\delta_{p_i}+4\pi\sum^{M}_{j=1}\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2, \end{equation} where…

Using a method mixing Mellin-Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary "$N$-point"…

For gauge field propagators, the asymptotic behavior is obtained in all directions of the complex $k^2$-plane, and for general, linear, covariant gauges. Asymptotically free theories are considered. Except for coefficients, the functional…

In this article, we prove some results concerning the truncated two-point function of the infinite-range Ising model above and below the critical temperature. More precisely, if the coupling constants are of the form $J_{x}= \psi(x)e^{…

We consider the asymptotic expansion of the Wright function \[W_{\lambda,\mu}(z)=\sum_{n=0}^\infty\frac{z^n}{n! \Gamma(\lambda n+\mu)}\qquad (\lambda>-1)\] for large (positive and negative) variable and large parameter $\mu$. The analysis…

We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln\Gamma(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer (1962), Spira (1971), and Hare (1997). We…

We derive the extended Ginzburg-Landau (GL) formalism for a clean s-wave two-band superconductor by employing a systematic expansion of the free-energy functional and the corresponding matrix gap equation in powers of the small deviation…

In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^{\alpha} e^{-N cx}$ supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in \mathbb{N}…

Based on the Gor'kov formalism for a clean s-wave superconductor, we develop an extended version of the single-band Ginzburg-Landau (GL) theory by means of a systematic expansion in the deviation from the critical temperature T_c, i.e.,…

The ultimate bound to the accuracy of phase estimates is often assumed to be given by the Heisenberg limit. Recent work seemed to indicate that this bound can be violated, yielding measurements with much higher accuracy than was previously…

We consider the asymptotics of the one-dimensional cubic nonlinear Schr\"odinger equation with an external potential $V$ that does not admit bound states. Assuming that $\jBra{x}^{2+}V(x) \in L^1$ and that $u$ is orthogonal to any…

For $p>2$, we consider the quasilinear equation $-\Delta_p u+|u|^{p-2}u=g(u)$ in the unit ball $B$ of $\mathbb R^N$, with homogeneous Neumann boundary conditions. The assumptions on $g$ are very mild and allow the nonlinearity to be…

In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ g(u), \quad {\rm in} \ \mathbb…

Let $X=\sum_{k=1}^\infty X_k \beta^{-k}$ be the base-$\beta$ expansion of a continuous random variable $X$ on the unit interval where $\beta$ is the positive solution to $\beta^n = 1 + \beta + \cdots + \beta^{n-1}$ for an integer $n\ge 2$…

We obtain two sequences of rational numbers which converge to the Euler-Gompertz constant. Denote by <f(x)> the integral of f(x)e^{-x} from 0 to infinity. Recall that the Euler-Gompertz constant \delta is <ln(x+1)>. Main idea. Let P_n(x) be…

Renormalized coupling constants g_{2k} that enter the critical equation of state and determine nonlinear susceptibilities of the system possess universal values g*_{2k} at the Curie point. They are calculated, along with the ratios R_{2k} =…

We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z|…

We bound the second eigenvalue of random $d$-regular graphs, for a wide range of degrees $d$, using a novel approach based on Fourier analysis. Let $G_{n, d}$ be a uniform random $d$-regular graph on $n$ vertices, and let $\lambda (G_{n,…