Generalized Bessel functions and Kapteyn series

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.

A gyrokinetic linearized exact (not model) Landau collision operator is derived by transforming the symmetric and conservative Landau form. The formulation obtains the velocity-space flux density and preserves the operator's conservative form as the divergence of this flux density. The operator contains both test-particle and field-particle contributions, and finite Larmor radius effects are evaluated in either Bessel function series or gyrophase integrals. While equivalent to the gyrokinetic Fokker-Planck form with Rosenbluth potentials [B. Li and D. R. Ernst, Phys. Rev. Lett. 106, 195002 (2011)10.1103/PhysRevLett.106.195002], the gyrokinetic conservative Landau form explicitly preserves the symmetry between test-particle and field-particle contributions, which underlies the conservation laws and the H theorem, and enables discretization with a finite-volume or spectral method to preserve the conservation properties numerically, independent of resolution. The form of the exact linearized field-particle terms differs from those of widely used model operators. We show the finite Larmor radius corrections to the field-particle terms in the exact linearized operator involve Bessel functions of all orders, while present model field-particle terms involve only the first two Bessel functions. This new symmetric and conservative formulation enables the gyrokinetic exact linearized Landau operator to be implemented in gyrokinetic turbulence codes for comparison with present model operators using similar numerical methods.

In the article, based on the equations of the theory of poroelastic bodies of Cowin-Nunziato, an axisymmetric contact problem of the interaction of a rigid stamp with a poroelastic cylinder of finite dimensions lying on a Winkler foundation is considered. It is assumed that the stamp base has a flat shape and a paraboloid of revolution, there is no friction in the contact zone. The cylinder rests on the Winkler foundation, and on the lateral surface of the cylinder there are no normal displacements and tangential stresses. The stamp moves vertically by a certain value using a given force. The problem posed, based on the representation of unknown displacements in the cylinder and the function, describing the change of the volume fraction of pores, in the form of series in Bessel functions, is reduced relative to the distribution function of contact stresses under the stamp to an integral equation of the first kind, the transform of the kernel of which is given in explicit form. To solve the integral equation, the collocation method is used, which allows finding contact stresses and the dependence of the value of the applied force on the displacement of the stamp. A comparative analysis of the studied values for different values of the Winkler foundation parameters, the porosity of the cylinder and its geometric parameters is carried out.

We present the solution of an axisymmetric problem of diffraction of electromagnetic fields on perfectly conducting conical shells. By assuming that the required solution is representable as a Kontorovich-Lebedev integral, we reduce the problem under investigation to a Fredholm integral equation of the first kind, whose solution has the form of a series in Bessel functions. For a radial electric dipole placed on the symmetry axis of the cone, the problem is reduced to two coupled infinite linear systems in the unknown expansion coefficients. We consider the case of low frequencies. The expression for the scattering diagram is obtained for the case of diffraction of plane waves by passing to the limit. The characteristics of the diffracted field and the possibility of control over certain parameters of the conical shell are studied numerically.

An exact expression for the probability of inversion of a large spin is established in the form of an asymptotic expansion in the series of Bessel functions with orders belonging to an arithmetic progression. Based on the new asymptotic expansion, a formula for the inversion time of the spin is derived.

Weighted divisor sums and Bessel function series, II

This paper presents the general theoretical analysis of three‐dimensional mechanical and thermal stresses for a short hollow cylinder made of functionally graded material. The steady‐state temperature, displacements, and stresses distributions are derived due to the general mechanical and thermal boundary conditions, as functions of radial, circumferential and longitudinal directions. The material properties vary continuously along the thickness direction according to power functions of radial direction. The temperature and Navier equations are solved analytically, using the generalized Bessel function and Fourier series. A direct method of solution of Navier equations is presented.

If F(x) is any function integrable and with an integrable square (in the sense of Lebesgue), then the two series (3) £<****, a* = --I F(x)x-»-idx, *-0 2îTt Jy (4) T,chpk(x), ck= fF(x)Pk(x)dx, *-0 Jy have on y (and henceW in the closed region \x \ = 1) essentially the same con-* Presented to the Society, September 9,1927; received by the editors in May, 1927.

This paper presents an alternative vector analysis of the electromagnetic (EM) fields radiated from electrically small thin square and rectangular loop antennas of arbitrary length 2a and width 2b . This method employs the dyadic Green's function (DGF) in the derivation of the EM radiated fields and thus makes the analysis general, compact and straightforward. Both near- and far-zones are considered so that the EM radiated fields are expressed in terms of the vector wave eigenfunctions. Not only the exact solution of the EM fields in the near and far zones outside the region (where r > √a2 + b2 ) is derived by use of spherical Hankel functions of the first kind, but also the closed series form of the EM fields radiated in the near zone inside the region 0 r < b is obtained in series of spherical Bessel functions of the first kind. The two regions between the radial distances b and √a2 + b2 which are defined as intermediate zones are characterized by both the spherical Bessel and Hankel functions of the first kind. Validity of the numerical results is discussed and clarified.

We extend a cell growth model with a deterministic cell growth rate and asymmetric division to include dispersion. This extension yields a second order functional differential equation, the solution of which is required to be a probability density function for the size of a cell (as measured by DNA content). We study the equation for certain choice of non constant coeffcients that correspond to dispersion, growth and splitting rates. This choice of coeffcients leads to a Bessel type operator, and it is shown that there is a unique probability distribution function that solves the equation. The solution is constructed using the Mellin transform and is given in terms of an infinite series of modified Bessel functions.

Expansion of the Operational Representation in the Series of Legendre and Bessel Functions

Anharmonic inter-atomic potential ∼|x|n , n > 1, has been used in molecular dynamics (MD) simulations of stress dynamics of FCC oriented crystal. The model of the chain of masses and springs is found as a convenient and accurate description of simulation results, with masses representing the crystallographic planes. The dynamics of oscillations of two planes is found analytically to be given by Euler’s beta functions, and its scaling with non-linearity parameter and amplitude of oscillations, or applied shear pressure is discussed on examples of time dependencies of displacements, velocities, and forces acting on masses (planes). The dynamics of stress penetration from the surface of the sample with multiply-planes (an anharmonic crystal) towards its interior is confirmed to be given exactly as a series of Bessel functions, when n = 2 (Schrödinger and Pater solutions). When n ≠ 2 the stress dynamics (wave propagation) in bulk material remains qualitatively of the same nature as in the harmonic case. In particular, results suggest that the quasi-linear relationship between frequency and the wave number is preserved. The speed of the transverse sound component, dependent on sound wave amplitude, is found to be a strongly decreasing function of n. The results are useful in the analysis of any MD simulations under pressure, as they help to understand the dynamics of pressure retarded effects, as well as help design the proper methodology of performing MD simulations in cases such as, for instance, studies of the dynamics of dislocations.

1. The determination of potentials and wave functions defined in regions bounded by the natural coordinate surfaces of a cylindrical polar coordinate system leads to eigenvalue problems connected with Bessel's differential equationWe shall consider the annulus 0 &lt; a ≤ r ≤ b and take first the boundary condition to be that of vanishing on both surfaces. It is then convenient to introduce the functionThis function will automatically satisfy the condition of vanishing for r = a. It will also vanish on r = b if u and k are related by the condition that ψ(u, k, b) = 0. If we regard u as being fixed and real then there will be an infinite number of zeros kn(u)(n = 1,2,…).

A certain series of Bessel functions-recently discussed by Lee (1988)-is an asymptotic expansion of an integral of a Bessel function. Here the asymptotic properties of the series are investigated in more detail, and it is shown that the series is not only asymptotic, but also convergent under suitable restrictions. For large positive real arguments finite numbers of terms of the series give good approximations to the integral, but the infinite sum is different from the integral.

Two methods for determining impact-force history on elastic plates