Computing the real parabolic cylinder functions U ( a , x ), V ( a , x )

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

Two-dimensional matrix parabolic cylinder beams

The basic parabolic cylinder (bpc) distributions of the firstBasic parabolic cylinder (bpc) distributionof the first kind and second kindsBasic parabolic cylinder (bpc) distributionof the second kind were derived by OgasawaraOgasawara (Commun Stat Theor Methods [17]) to obtain the absolute momentsMomentsabsolute of the normal variable underStripe truncation stripe (zebraicTruncationzebraic or tigerishTruncationtigerish) truncationTruncationstripe. The bpc distribution of the third kindBasic parabolic cylinder (bpc) distributionof the third kind or simply bpc is introduced as a general bpc, whose special cases are the bpc’s of the first and second kinds. TheWeighted or incomplete parabolic cylinder function weighted or incomplete parabolic cylinder functionParabolic cylinder functionincomplete is obtained using theWeighted or incomplete Kummer confluent hypergeometric function weighted or incomplete Kummer confluent hypergeometric functionKummer confluent hypergeometric function given by OgasawaraOgasawara [J Multivar Anal [18]) to have the absolute momentsMomentsabsolute. The multivariate bpc distributionMultivariate basic parabolic cylinder (bpc) distribution is also derived to obtain the absolute momentsMomentsabsolute of the normal vectorNormalvector underTruncationsectional sectional truncationSectional truncation, which is the multivariate version ofTruncationstripe stripe truncationStripe truncation.

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

In this paper, we study the uniform asymptotics of the Meixner‐Pollaczek polynomials with varying parameter as , where A > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for z in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms of elementary functions only. Our approach is based on the steepest descent method for oscillatory Riemann‐Hilbert problems first introduced by Deift and Zhou [1].

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate J v ( x ) ( | x | small ) {J_v}(x)\;(|x|\;{\text {small}}) , e x K v ( x ) ( Re ⁡ x > 0 , | x | large ) {e^x}{K_v}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}}) , and the modulus and phase of H v ( 1 ) ( x ) ( Re ⁡ x > 0 , | x | large ) H_v^{(1)}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}}) . The first converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for K v + 1 ( x ) / K v ( x ) {K_{v + 1}}(x)/{K_v}(x) the convergence of which increases with | x | |x| . The procedures were tested numerically both for integer and fractional values of v , and for real and complex x .

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

: The usual power series representations for hypergeometric functions in two variables have a limited range of validity. In particular, they are of little use when the magnitude of one of the variables becomes very large. Using a Barnes-type integral representation and the concept of analytic continuation, the region of utility is extended to the desired domain. The poles that occur in the Barnes-type integral are assumed to be simple. Thus, explicit asymptotic expansions are obtained for each of the fourteen hypergeometric functions that belong to this class. Hypergeometric functions, Functions with a large argument, Functions of two variables, Contour integration, Power series expansions.

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.

We consider the asymptotic behavior of solutions of the first -difference Painlevé equation in the limits and . Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behavior associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behavior is described by a power series expression, and find that the Stokes curves are described by curves known as -spirals.

We show that the partial sums of the power series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so, we obtain information about the zeros of the partial sums. We will only assume that these entire functions have a certain asymptotic behavior at infinity. With this information, we will partially verify for this class of functions a conjecture on the location of the zeros of their partial sums known as the Saff–Varga width conjecture. Numerical results and figures are included to illustrate the results obtained for several well-known functions including the Airy functions and the parabolic cylinder functions.

In this paper, inspired by V. A. Borovikov’s pioneering work, we study a wave diffracted by a convex cone of arbitrary cross section. Our analysis is focused on the shadow boundary of the wave reflected by the cone surface, where geometric optics yields nonphysical weak singularity. The uniform asymptotic expansion constructed here, which is expressed in terms of the parabolic cylinder functions, is regular at the geometric shadow boundary. Outside the penumbra region, it perfectly matches the ray expansions of the reflected wave, yielding in addition a regular spherical wave, would-be scattered by the cone vertex.

23 - Unified Treatment of Ideal Fermi, Bose, and Classical Gases

In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points --- two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given.