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Exact solvability of one-dimensional quantum-mechanical potentials has extensively been studied and constructed, yet there remain other interaction models whose wave functions are given by special functions. In this paper, we discuss an exactly solvable Schrödinger equation where the eigenfunctions are expressed in terms of the Bessel functions of purely imaginary order. Our potential is defined by a piecewise analytic function, and possesses Bessel-function solvability. We compute the whole bound-state spectra as well as the scattering solutions.

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as $d$-dimensional balls (with $d=1,2,3$), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

A new computational procedure is offered to provide simple, accurate and flexible methods for using modern computers to give numerical evaluations of the various Bessel functions. The trapezoidal rule, applied to suitable integral representations, may become the method of choice for evaluation of the many special functions of mathematical physics.

A fully canonical quantization of electromagnetic field in the presence of a bi-anisotropic absorbing magneto-dielectric cylindrical shell is provided. The mode expansions of the dynamical quantum fields, contained in the theory, is achieved and the ladder operators of the system are introduced. Using the Frobenius’s series technique, the Maxwell’s equations in the presence of the bi-anisotropic absorbing magneto-dielectric cylindrical shell are solved and the space–time dependence of the quantized electromagnetic field is obtained. Applying the conservation principle of the angular momentum, the net quantum vacuum torque exerted on the bi-anisotropic absorbing magneto-dielectric cylindrical shell is calculated. The net quantum vacuum torque exerted on the cylindrical shell is calculated in the vacuum state and the thermal state of the system. The quantum vacuum torque on the cylindrical shell identically vanishes when the bi-anisotropic absorbing magneto-dielectric cylindrical shell is converted to an isotropic one.

The Bessel equation is shown to be equivalent, under suitable transformations, to a system of two damped/amplified parametric oscillator equations, which have been used in the study of inflationary models of the Universe, thermal field theories and Chern–Simons gauge theories. The breakdown of loop-antiloop symmetry due to group contraction manifests itself as breaking of time-reversal symmetry. The relation between some infinite dimensional loop-algebras, such as the Virasoro-like algebra, and the Euclidean algebras e(2) and e(3) is also analyzed.

We consider three-dimensional inverse scattering with fixed energy for which the spherically symmetrical potential is nonvanishing only in a ball. We give exact upper and lower bounds for the phase shifts. We provide a variational formula for the Weyl–Titchmarsh m-function of the one-dimensional Schrödinger operator defined on the half-line.

The spreading of the position and momentum probability distributions for the stable free oscillations of a circular membrane of radius l is analyzed by means of the associated Boltzmann–Shannon information entropies in the correspondence principle limit (n → ∞, m fixed), where the numbers (n, m), n ∈ ℕ and m ∈ ℤ, uniquely characterize an oscillation of this two-dimensional system. This is done by solving the short-wave asymptotics of the physical entropies in the two complementary spaces, which boils down to the calculation of the asymptotic behavior of certain entropic integrals of Bessel functions. It is rigorously shown that the position and momentum entropies behave as 2ln(l) + ln(4π) - 2 and ln(n) - 2ln(l) + ln(2π³) when n → ∞, respectively. So the total entropy sum has a logarithmic dependence on n and it does not depend on the membrane radius. The former indicates that the ordering of short-wavelength oscillations is exactly identical for the entropic sum and the single-particle energy. The latter holds for all oscillations of the membrane because of the uniform scaling invariance of the entropy sum.

In this work, we extend the analytic treatment of Bessel functions of large order and/or argument. We examine uniform asymptotic Bessel function expansions and show their accuracy and range of validity. Such situations arise in a variety of applications, particularly the Fourier transform (FT) of the gravitational wave (GW) signal from a pulsar, global parameter space correlations of a coherent matched filtering search for continuous GWs from isolated neutron stars and tomographic reconstruction of GW LISA sources. The uniform expansion we consider here is found to be valid in the entire range of the argument.

Several existence and nonexistence results are known for positive solutions u ∈ D^1,2(ℝ^N) ∩ L²(ℝ^N, ∣x∣^-αdx) ∩ L^p(ℝ^N) to the equation

In this paper, the elastic lateral-torsional buckling of tapered cantilever strip beams acted by a tip load is investigated. Analytical solutions of the buckling problem are derived for cantilevers with a linearly varying depth. It is shown that the corresponding governing differential equation is a particular case of the general confluent equation, whose solution is given in terms of confluent hypergeometric functions. The buckling load corresponds to the lowest positive root of a 4×4 characteristic determinant arising from the introduction of the general solution of the differential equation into the boundary conditions. Finally, one discusses the optimization of the cantilever geometrical shape (for constant volume), as far as its resistance to lateral-torsional buckling is concerned.

The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to (convergent) Laurent series. It is thought that these more general differential equations can be used as an alternative description of some special functions.

There are now several ways to derive an asymptotic expansion for , as n → ∞, which holds uniformly for . One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for . Next, we give an estimate for the error term which is of the same order as the first neglected term.

Some integral representations involving Legendre functions are given. Some are related to Bessel functions and their products. Others are concerned with zeros of Legendre functions and may provide a suitable basis for computational purposes.

The classical modal expansion for the scattered field of a plane wave from a circular dielectric cylinder is studied. A new uniform asymptotic approximation is presented for the late coefficients in this expansion, in the case of a fixed relative dielectric constant ε_r, both real and complex. These new approximations for the mode values are not based on the scattering matrix but rather the classical WKBJ approximations for the Bessel functions, and are valid for the entire region exterior to the cylinder, including the transition region. Furthermore, a precise asymptotic form for the location of a certain critical Regge pole is obtained. It is shown that this pole can lead to at least one dramatic resonant modal term at certain critical values, and the exponential nature of the mode in question is determined explicitly. This is followed by an extension to complex values of ε_r with new uniform asymptotic approximations for the modes also being obtained, and these in turn demonstrate a heavy damping of the resonant mode.

The function is studied. By employing uniform asymptotic approximations for Bessel functions, as well as Nicholson's integral for and a related integral, uniform asymptotic approximations for M_ν(x) are obtained for x → ∞, which taken together are uniformly valid for -∞ < ν < ∞. From these approximations, it follows that M_ν(x) is a slowly varying function of ν for large x, a result which has ramifications in a certain quasi non-uniqueness in the scattered field of a dielectric circular cylinder.

It is well known that the scattered field of a z polarized plane wave incident on a dielectric circular cylinder can be expanded as an infinite series involving Hankel functions. From numerical calculations of this expansion, Lam and Yedlin [5] observed that the mean square measure, over all space, of the difference of the scattered fields from two or more distinct values of the dielectric constant of the cylinder can take very small values, thereby almost contradicting the uniqueness property. We investigate this phenomenon rigorously using uniform asymptotic expansions of Bessel functions, and from our analysis we determine the spurious values of the dielectric constant which lead to this quasi-nonuniqueness.

The solution of the Fokker–Planck equation for exponential Brownian functionals usually involves spectral expansions that are difficult to compute explicitly. In this paper, we propose a direct solution based on heat kernels and a new integral representation for the square modulus of the Gamma function. A financial application to bond pricing in the Dothan model is also presented.

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation

We consider the second-order linear differential equation

The aim of this paper is to derive new representations for the Hankel and Bessel functions, exploiting the reformulation of the method of steepest descents by Berry and Howls [Hyperasymptotics for integrals with saddles, Proc. R. Soc. Lond. A434 (1991) 657–675]. Using these representations, we obtain a number of properties of the large-order asymptotic expansions of the Hankel and Bessel functions due to Debye, including explicit and numerically computable error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.