Relations between Weber's Parabolic Cylinder Functions and the k-Functions

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.

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.

Two-dimensional matrix parabolic cylinder beams

6–7 - Definite Integrals of Special Functions

We present new electromagnetic plasma wavepacket solutions that propagate in one time and one space coordinates. Unlike the usual plane wave solution, which is written in terms of separation of variables, all our solutions are defined along the light-cone coordinates. This allows us to find several new solutions whose functionality properties rely on the conditions imposed on the choice for their light-cone coordinates dependence. The presented wavepacket solutions are constructed in terms of multiplications of Airy functions, Parabolic cylinder functions, Mathieu functions, or Bessel functions. We thoroughly analyze the case of a double Airy solution, which exhibits new electromagnetic properties, as a defined wavefront, and velocity faster than the electromagnetic plane wave counterpart solution. It is also mentioned how more general structured wavepackets can be constructed from these new solutions.

A study of linear systems with time varying coefficients by the methods of bessel and weber functions

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.

Purpose: This study investigates obliquely inclined broad proton beams of energy in the therapeutic region. Its purpose is to provide analytic modeling of the depth-dose distribution of such beams Method and Materials: A new theory has been developed based on the work of Bortfeld (Med. Phys. 24, 2024–2033, December 1997) for normally incident proton beams. Parabolic cylinder functions are used, and for each configuration (incident angle, initial energy, material penetrated) it is assumed that the r.m.s. lateral spread of a pencil beam (σx) is constant at all depths. MCNPX Monte Carlo simulations (10 or 20 million histories) are used to test and verify the accuracy of the theory in modeling the broad-beam depth dose. This comparison has been made for a total of eighty configurations: a homogeneous phantom composed of water, bone, aluminum, or copper; an initial energy of 50, 100, 150, 200, or 250 MeV; and an angle of incidence of 0°, 15°, 30°, or 45°. Results: There is very close agreement between theory and Monte Carlo calculations, for water and bone phantoms; discrepancies are more pronounced for aluminum and, especially, for copper. Moreover, for a particular phantom material the optimal value of σx depends little on the angle of incidence. Furthermore, for each phantom material the optimal value of σx very closely follows a power law as a function of incident energy or of initial range of the protons. Conclusion: The excellent results are rather surprising, given the gross approximation of σx being constant in depth for each configuration. The power-law dependence of σx as a function of incident energy or of initial range also comes as surprise. Further investigation is called for, in understanding this high accuracy, but presumably it occurs because the depth-dose curve varies only slowly before the Bragg peak region is reached.

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.

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd, or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.

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.

A linear dynamical system is considered whose normal frequencies and normal modes come into close coincidence. The case when both modes have positive energy has been discussed by Grimshaw and Allen (1979). Here the case when one mode has positive energy and the other mode has negative energy is discussed. Coupled equations are derived and solved exactly using parabolic cylinder functions. It is found that the action in both modes grows during the coupling.

Amperometric response of a rectangular channel electrode

Indoors, GNSS signal encounters severe multipath power loss and fading which leads to significant signal degradation of the amplitude and phase to perform GPS (or GNSS) signal acquisition. To overcome these effects, piling up received GPS (or GNSS) data is a traditional (or conventional) method; however, it exhibits two shortcomings (drawbacks or limitations): instable detection performance, such as the probability of false alarm (PFA) fluctuates due to changes of the signal-to-noise ratio (SNR); and elongated acquisition time caused by extended accumulation duration and over repetitive FA occurrence (or “penalty”). To overcome the first shortcoming, an adaptive structure is employed in GNSS signal acquisition that enables constant FA rate (CFAR) criteria to guarantee a stable detection performance on the GNSS signal acquisition. To overcome the second shortcoming, an adaptive determination on accumulation length is employed to minimize the accumulation duration; therefore, a double-dwell structure (DDS) is used to reduce the processing time “penalty” caused by FA. Simulation results illustrate that an adaptive, stable detection implementation and DDS reduce the average acquisition time by almost fifty percent (or by half or a factor of two). Other important significant unique accomplishments found in this paper are as follows: Dr. Progri for the first time provides the closed form expressions of generalized modified Bessel function distributions of the cumulative distribution functions (cdf) of the first and second kinds and for the first time introduces the very powerful utility of the Kampe de Feriet function and parabolic cylinder function in the navigation community. Index Terms—Adaptive GNSS signal acquisition; weak GNSS signal; constant FA rate (CFAR); DDS; generalized modified Bessel function distribution; modified Bessel function; Kampe de Feriet function; parabolic cylinder function.

We present an alternative to conventional finite difference solutions to the transport equations for electrons and holes in nondegenerate semiconductor devices. The equations are solved using piecewise analytic methods in which the electric field is approximated by a linear function in a series of relatively large subintervals, and the continuity and current density equation is solved by transforming it into Weber’s equation. Carrier concentrations are expressed as a linear combination of Weber parabolic cylinder functions in each interval. Selfconsistency is obtained by iterative corrections with Gummel’s method. Computational results are presented for an abrupt P-N+ junction. It is believed that this method is particularly efficient for moderate bias conditions and especially in modeling heterojunction solar photovoltaic devices.