Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a Cauchy integral, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem,...

One particular strength of the boundary element method is that it allows for a high-order pointwise approximation of the solution of the related partial differential equation via the representation formula. However, the high-order convergence and hence accuracy usually suffers from singularities of the Cauchy data. We propose two adaptive mesh-refining algorithms and prove their quasi-optimal c...

We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the half-line (0,∞) and the interval (−1, 1). This process is related to the square root of one-dimensional Laplacian A = − √ − d dx with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula ...

We consider the estimation of the number of labelled simple graphs with degree sequence d1, d2, . . . , dn by using an n-dimensional Cauchy integral. For suffiently small and any c > 2 3 , an asymptotic formula is obtained when |di − d| < n for all i and d = d(n) satisfies min{d, n − d − 1} ≥ cn/ log n as n → ∞. These conditions include the degree sequences of almost all graphs, so our result g...

Taylor’s polynomial is a central tool in any elementary course in mathematical analysis. Nowadays, its importance is centred on its applications, for instance, to asymptotic analysis or to obtain satisfactory numerical or integral inequalities see, e.g., 1–5 . The core of these results comes from manipulations on the explicit formula of the remainder, that is, the error estimation when consider...

The absolute refractive index of liquid sulphur dioxide (SO2) was determined with an uncertainty of ±0.0007 with the help of a Michelson interferometer and a computer-controlled rotatory stage supporting a rectangular fused silica cell, containing the liquid. The measurements were performed at temperatures around 20 ◦C for the laser line wavelengths 632.8, 514.5 and 488.0 nm. The experimental d...

This is a continuation of a series of papers [FL1, FL2, FL3], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we continue to study the quaternionic analogues of Cauchy’s formula for the second order pole. These quaternionic analogues are closely related to regularization of infinities of vacuum ...