Convergence versus correspondence for sequences of rational functions

AbstractLet f be meromorphic in the plane and analytic at 0. Then its diagonal sequence {[n/n]}∞n = 1 of Padé approximants need not converge pointwise. We ask whether by reducing the order of contact (or correspondence) of [n/n] with f at 0, namely 2n + 1, we can ensure locally uniform convergence. In particular, we show that there exist rational functions Rn of type (n, n), n ≥ 1, and a sequence of positive integers {ln}∞n = 1 with limit ∞, depending on f, such that Rn has contact of order n + ln + 1 with f at 0, and which converge locally uniformly to f. Moreover, for any given sequence {ln}∞n = 1, there exists an entire f for which order of contact higher than n + ln is incompatible with convergence

On the Coulomb-Sturmian matrix elements of the Coulomb Green's operator

The two-body Coulomb Hamiltonian, when calculated in Coulomb-Sturmian basis, has an infinite symmetric tridiagonal form, also known as Jacobi matrix form. This Jacobi matrix structure involves a continued fraction representation for the inverse of the Green's matrix. The continued fraction can be transformed to a ratio of two $_{2}F_{1}$ hypergeometric functions. From this result we find an exact analytic formula for the matrix elements of the Green's operator of the Coulomb Hamiltonian.Comment: 8 page

The two-level atom laser: analytical results and the laser transition

The problem of the two-level atom laser is studied analytically. The steady-state solution is expressed as a continued fraction, and allows for accurate approximation by rational functions. Moreover, we show that the abrupt change observed in the pump dependence of the steady-state population is directly connected with the transition to the lasing regime. The condition for a sharp transition to Poissonian statistics is expressed as a scaling limit of vanishing cavity loss and light-matter coupling, $\kappa \to 0$, $g \to 0$, such that $g^2/\kappa$ stays finite and $g^2/\kappa > 2 \gamma$, where $\gamma$ is the rate of atomic losses. The same scaling procedure is also shown to describe a similar change to Poisson distribution in the Scully-Lamb laser model too, suggesting that the low-$\kappa$, low-$g$ asymptotics is of a more general significance for the laser transition.Comment: 23 pages, 3 figures. Extended discussion of the paper aim (in the Introduction) and of the results (Conclusions and Discussion). Results unchange

Ramanujan and Extensions and Contractions of Continued Fractions

If a continued fraction $K_{n=1}^{\infty} a_{n}/b_{n}$ is known to converge but its limit is not easy to determine, it may be easier to use an extension of $K_{n=1}^{\infty}a_{n}/b_{n}$ to find the limit. By an extension of $K_{n=1}^{\infty} a_{n}/b_{n}$ we mean a continued fraction $K_{n=1}^{\infty} c_{n}/d_{n}$ whose odd or even part is $K_{n=1}^{\infty} a_{n}/b_{n}$. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if $K_{n=1}^{\infty}a_{n}/b_{n}$ is the even part); (ii) Find the limit of the other contraction and show that the odd and even parts of the extension tend to the same limit. We apply these ideas to derive new proofs of certain continued fraction identities of Ramanujan and to prove a generalization of an identity involving the Rogers-Ramanujan continued fraction, which was conjectured by Blecksmith and Brillhart.Comment: 16 page

On the possible role of cusp/cleft precipitation in the formation of polar-cap patches

International audienceThe work describes experimental observations of enhancements in the electron density of the ionospheric F-region created by cusp/cleft particle precipitation at the dayside entry to the polar-cap convection flow. Measurements by meridian scanning photometer and all-sky camera of optical red-line emissions from aurora are used to identify latitudinally narrow bands of soft-particle precipitation responsible for structured enhancements in electron density determined from images obtained by radio tomography. Two examples are discussed in which the electron density features with size scales and magnitudes commensurate with those of patches are shown to be formed by precipitation at the entry region to the anti-sunward flow. In one case the spectrum of the incoming particles results in ionisation being created, for the most part below 250 km, so that the patch will persist only for minutes after convecting away from the auroral source region. However in a second example, at a time when the plasma density of the solar wind was particularly high, a substantial part of the particle-induced enhancement formed above 250 km. It is suggested that, with the reduced recombination loss in the upper F-region, this structure will retain form as a patch during passage in the anti-sunward flow across the polar cap