The Basic Parabolic Cylinder Distribution and Its Multivariate Extension

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.

Some remarks on the location of non-asymptotic zeros of Whittaker and Kummer hypergeometric functions

The propagation of circularly symmetrical mode in an infinitely long thin-wire helix, wound on a latching ferrite cylinder, magnetised azimuthally to remanence and surrounded by homogeneous, isotropic, lossless dielectric and perfectly conducting metallic shield is studied analytically. The characteristic equation of the system, involving the normalized phase constant, structure geometry, parameters of both media and the frequency is derived in terms of modified Bessel functions and Kummer confluent hypergeometric functions. The nonreciprocal phase characteristics of the structure, assuming Marconi latching ferrite L1A and thermal conducting dielectrics are presented in normalized form for opposite directions of remanent flux, showing application of the system as practical remanent latching phaser.

We explore the application of fundamental solutions, constructed as irreducible linear combinations with variable coefficients of two Kummer confluent hypergeometric functions, to solve the one-dimensional stationary Schrödinger equation. This approach offers a new technique for tackling this equation, particularly well-suited for singular potentials exhibiting non-monotonic behavior, a challenge for traditional methods. We focus on a specific family of four-parameter potential wells defined on the positive semi-axis using the Lambert-W function. For these wells, the potential energy increases toward the origin but tends to zero at infinity. We present solutions in the form of linear combinations of Kummer functions and their derivatives, along with power functions and exponents. We then explore two specific well types within this family, analyzing the energy spectrum of the quantum system for each case.

The borderlines of the area in which the circular waveguide, entirely filled with azimuthally magnetized ferrite, works as a phase shifter for the normal TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">01</sub> mode, are determined. For the purpose the special features of the cut-off regime of the configuration and of its limiting one, observed at higher frequencies in which the propagation stops, provided the magnetic bias is negative, are used. For a fixed numerical equivalent of the normalized in an appropriate way guide radius, the off-diagonal ferrite permeability tensor element and the phase constant of the wave are counted by the roots of the structure's characteristic equation, derived through the Kummer confluent hypergeometric function. This is done repeatedly for a varying negative (positive) value of the imaginary part of its complex first parameter. The process continues, until the computed value of the element coincides with that of the same, corresponding to the cut-off state for the radius chosen (to the limiting one, if the latter is greater than certain number) with a prescribed accuracy. The relevant calculated numerical equivalent of the phase constant is accepted as the one, searched for. The differential phase shift equals the constant mentioned in the first case, whereas in the second one, it is found from it by a simple arithmetic. Changing the normalized radius, the boundaries of the domain studied are traced. The outcomes are presented graphically in a normalized form.

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

6–7 - Definite Integrals of Special Functions

Power series for the parabolic cylinder function, Dv(z), are derived from known addition theorems, thus enabling a more compact and efficient expression of the function. One of the expansions is applied to find an asymptotic approximation for the function as mod v mod to infinity with mod arg(-v) mod ( pi . Next, a large-argument (z) asymptotic addition theorem is derived from an integral representation of Dv(z). Before extending this type of summation theorem to the two general confluent hypergeometric functions, motivating mathematical applications to three integration problems are given as examples of the practical usefulness of the formulae.

This thesis is concerned with the mathematical study of steady two-dimensional Marangoni flow problems in both planar and radial geometries. By exploiting the recently discovered connection between the forced complex Burgers equation and Marangoni flows in a viscous fluid region, first discovered by D. G. Crowdy, it shall be shown that three physically distinct problems can each be reduced to a single forced complex Burgers equation, satisfied by a complex function that is analytic in the fluid domain. The contribution of this thesis is to demonstrate that, in all three cases, steady equilibrium solutions to these equations are given by special function solutions of certain well-known ordinary differential equations. Specifically, the parabolic cylinder and doubly-confluent Heun equations are considered. The first physical problem considers a localised concentration of insoluble surfactant occupying an infinite flat interface between a deep viscous fluid region experiencing a linear extensional flow and a constant pressure region. This scenario is then generalised to a two-phase analogue, where the surfactant is now soluble to one of the fluids. In both cases, solutions are found in terms of logarithmic derivatives of the parabolic cylinder function. The solutions and their singularities are analysed in the limit of vanishing surface diffusion using asymptotic techniques and Liouville-Green analysis. These results are then discussed in the context of the Stokes phenomenon for nonlinear ODEs. The third and final problem considered in this thesis is that of a bubble laden with insoluble surfactant translating steadily through a viscous fluid region that is experiencing a linear temperature gradient. Equilibrium solutions are found in terms of a logarithmic derivative of a solution of the doubly-confluent Heun equation, which is determined by a monodromy condition imposed on the equilibrium solution. A semi-explicit expression for the bubble speed is derived.

In this paper an advanced anti-jam indoor adaptive GNSS signal acquisition and tracking algorithm is considered. Initially, we were able to determine that double-dwell structure (DDS) reduces the processing time penalty caused by false alarm. Nevertheless, the double-dwell structure is still vulnerable to interference and jamming. In order to determine a suitable advanced anti-jam indoor adaptive GNSS signal acquisition and tracking algorithm for DDS we first perform the Bayesian parameter estimation; i.e., we analytically compute the posterior Bayes probability density function (pdf) and cumulative distribution function (cdf) by applying the Bayes theorem in three steps. First, we compute the complex signal distribution and complex matrix variate signal distribution. This is an original new result never published before. Second, we provide an introduction of the complex Bessel or parabolic function interference distribution due to earlier assumptions of the DDS interference distribution models. We further explain that the process that is required to produce Complex Matrix Variate Bessel Interference Distribution or Complex Matrix Variate Parabolic Function Interference Distribution is an extremely daunting task let alone the Complex Matrix Variate DDS Bayesian posterior density it would seem at the moment nearly an impossible task because it may require the computation of functions such as Kampe de Feriet function and Jack functions of matrix arguments. Third, instead of performing the second step, we relax the assumption of interference to normal distribution. In both the scalar case and complex matrix variate cases we observe that the complex matrix variate Bayesian posterior pdf or cdf is invariant of the observation data or is identical to the prior complex matrix variate signal distribution model. This is an original new and very powerful result never published before. Why this result is so powerful is because up until now we never had a complete theoretical validation of our GNSS receiver design based on either autocorrelation or cross-correlation properties since, the complex matrix variate Bayesian posterior pdf or cdf is invariant of the observation data or is identical to the prior complex matrix variate signal distribution model. Future simulation results will illustrate that an anti-jam indoor adaptive DDS stable detection structure reduces by half the average acquisition time and significantly outperform its predecessor against interference and jamming. Moreover, future simulation results will show further advancements of the Giftet Inc. MATLAB library capability to perform advanced numerical computations based on closed form expressions of the generalized Bessel function distributions and generalized parabolic cylinder function distribution models via Kampe de Feriet function and Jack functions.

Two-dimensional matrix parabolic cylinder beams

An exact analytical solution for the universal probability distribution of the order parameter fluctuations as well as for the universal statistical and thermodynamic functions of an ideal gas in the whole critical region of Bose–Einstein condensation (BEC) is obtained. A universal constraint nonlinearity is found that is responsible for all nontrivial critical phenomena of the BEC phase transition. Simple analytical approximations, which describe the universal structure of the critical region in terms of confluent hypergeometric or parabolic cylinder functions, as well as asymptotics of the exact solution are derived. The results for the order parameter, all higher order moments of BEC fluctuations and thermodynamic quantities, including specific heat, perfectly match the known asymptotics outside critical region as well as the phenomenological renormalization-group ansatz with known critical exponents in the close vicinity of the critical point. Thus, a full analytical solution to a long-standing problem of finding a universal structure of the λ point for BEC in an ideal gas is found.

Examples of the application of confluent hypergeometric functions in miscellaneous areas of theoretical physics are presented. Emphasis is placed on the use of Kummer function in the field of microwaves: the cases of normal and slow rotationally symmetric TE modes propagation in the azimuthally magnetized circular ferrite waveguide are considered. Some of the properties of function mentioned in the complex (real) domain, of importance in the solution of boundary-value problem stated for normal (slow) waves, are established. A theorem for the identity of purely imaginary and real zeros of the complex resp. real Kummer function for certain parameters, is proved numerically. The terms for wave transmission are obtained as four bilaterally open intervals of variation of quantities, specifying the fields. It turns out that the normal (slow) modes may exist in one (two) region(s). The theoretically predicted phase curves for the first waves of the two TE sets examined show that the structure explored is suitable for ferrite control components design.

Methods for the computation of real parabolic cylinder functions U ( a , x ), and V ( a , x ) and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing parabolic cylinder functions for unrestricted values of the order a and the variable x except for the overflow/underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow/underflow limitations. In an accompanying article we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.