Modified Bessel Functions Research Articles

Steady-periodic thermal stresses in an infinite hollow compound cylinder

Closed-form expressions for temperatures and elastic thermal stresses in an infinite hollow two-material compound cylinder subject to steady-periodic sinusoidal ambient temperatures are presented, taking finite heat transfer coefficients into account. The fluctuating temperatures and stresses in the cylinder are expressed in terms of complex-valued modified Bessel functions under the assumption of temperature-independent material properties. Transient thermal loads composed of linear temperature ramps are expressed as Fourier series, and studied by the presented method. In order to verify the analytic results, examples involving sinusoidal and linear loads are examined by finite element calculations.

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kiν(z) with Respect to Order

The paper presents the derivation of the asymptotic behavior of $\nu$-zeros of the modified Bessel function of imaginary order $K_{{\rm i}\nu}(z)$. This derivation is based on the quasiclassical treatment of the exponential potential on the positive half axis. The asymptotic expression for the $\nu$-zeros (zeros with respect to order) contains the Lambert $W$ function, which is readily available in most computer algebra systems and numerical software packages. The use of this function provides much higher accuracy of the estimation comparing to known relations containing the logarithm, which is just the leading term of $W(x)$ at large $x$. Our result ensures accuracies sufficient for practical applications.

Bounds for the generalized Marcum function of the second kind

In this paper, we focus on the generalized Marcum function of the second kind of order nu >0, defined by Rν(a,b)=ca,νaν-1∫b∞tνe-t2+a22Kν-1(at)dt,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} R_{\\nu }(a,b)=\\frac{c_{a,\\nu }}{a^{\\nu -1}} \\int _b ^ {\\infty } t^{\\nu } e^{-\\frac{t^2+a^2}{2}}K_{\\nu -1}(at)\\mathrm{d}t, \\end{aligned}$$\\end{document}where a>0, bge 0,K_{nu } stands for the modified Bessel function of the second kind, and c_{a,nu } is a constant depending on a and nu such that R_{nu }(a,0)=1. Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones.

Second quantization and the spectral action

We consider both the bosonic and fermionic second quantization of spectral triples in the presence of a chemical potential. We show that the von Neumann entropy and the average energy of the Gibbs state defined by the bosonic and fermionic grand partition function can be expressed as spectral actions. It turns out that all spectral action coefficients can be given in terms of the modified Bessel functions. In the fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential μ approaches 0, can be expressed in terms of the Riemann zeta function. This recovers a result of Chamseddine-Connes-van Suijlekom.

Bound states and energy eigenvalues of a radial screened Coulomb potential

We analyze bound states and other properties of solutions of a radial Schrödinger equation with a new screened Coulomb potential. In particular, we employ hypervirial relations to obtain eigen-energies for a Hydrogen atom with this potential. Additionally, we appeal to a sharp estimate for a modified Bessel function to estimate the ground state energy of such a system. Finally, when the angular quantum number ℓ ≠ 0, we obtain evidence for a critical screening parameter, above which bound states cease to exist.

Dynamic response of a hollow cylinder subjected to thermal shock considering scale effect and memory dependent effect

Based on the generalized nonlocal thermoelastic theory and the memory-dependent differential (MMD) theory, the dynamic response of the inner surface of an infinite hollow cylinder under thermal shock is investigated. The heat transfer coefficient varies linearly with the temperature, the outer surface of the hollow cylinder is insulated, and there is traction-free on the inner and outer surfaces. The governing equation of the problem is established, and the Laplace transform is used to solve the governing equation, and the distribution law of physical quantity is expressed by the modified Bessel function. In the numerical calculation, the influence of non-local parameters is firstly studied. Second, the influence of different thermal conductivity on various physical quantities is studied. Finally, the effects of time delay factor and kernel function on physical quantities are studied. The results show that the non-local parameters have little effect on temperature, but have significant effect on displacement and stress. The change of thermal conductivity has obvious influence on temperature, displacement, and stress. The hysteresis factor affects the extremum of a physical quantity.

Asymptotics of the hitting probability for a small sphere and a two dimensional Brownian motion with discontinuous anisotropic drift

We provide an approximation of the hitting probability for a small sphere for the following two dimensional process: In x-direction it is just a Brownian motion with positive constant drift, whereas in y-direction the process Yt is a Brownian motion with drift given by a negative constant times the sign of Yt. This process can be seen as the solution of a certain stochastic optimal control problem. It turns out that the approximating function can be expressed as the sum of a term involving a modified Bessel function and an ordinary Lebesgue integral.

Узагальнення моделі Голанда і Рейсснера на випадок осьової симетрії

The purpose of this work is to create a mathematical model of the stress state of overlapped circular axisymmetric adhesive joints and to build an appropriate analytical solution to the problem. To solve the problem, a simplified model of the adhesive bond of two overlapped plates is proposed. The simplification is that the movement of the layers depends only on the radial coordinate and does not depend on the angular one. The model is a generalization of the classical model of the connection of Holland and Reissner in the case of axial symmetry. The stresses are considered to be evenly distributed over the thickness of the layers, and the adhesive layer works only on the shift. These simplifications allowed us to obtain an analytical solution to the studied problem. The problem of the stress state of the adhesive bond of two plates is solved, one of which is weakened by a round hole, and the other is a round plate concentric with the hole. A load is applied to the plate weakened by a round hole. The discussed area is divided into three parts: the area of bonding, as well as areas inside and outside the bonding. In the field of bonding, the problem is reduced to third- and fourth-order differential equations concerning tangent and normal stresses, respectively, the solutions of which are constructed as linear combinations of Bessel functions of the first and second genera and modified Bessel functions of the first and second genera. Using the found tangential and normal stresses, we obtain linear inhomogeneous Euler differential equations concerning longitudinal and transverse displacements. The solution of the obtained equations is also constructed using Bessel functions. Outside the area of bonding, displacements are described by the equations of bending of round plates in the absence of shear forces. Boundary conditions are met exactly. The satisfaction of marginal conditions, as well as boundary conditions, leads to a system of linear equations concerning the unknown coefficients of the obtained solutions. The model problem is solved and the numerical results are compared with the results of calculations performed by using the finite element method. It is shown that the proposed model has sufficient accuracy for engineering problems and can be used to solve problems of the design of aerospace structures.

K-combined random fields: Basic properties and stochastic orderings

This paper introduces the K-combined vector random field, whose finite-dimensional characteristic functions are made up of certain power functions and whose finite-dimensional density functions are comprised of the modified Bessel functions of the second type. It is an elliptically contoured vector random field, contains K-differenced vector random field of Alsultan and Ma (2019) as a special case, and possesses all orders of moments. It is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. With various selection of its parameters, its finite-dimensional distributions may have heavy tails or thin tails, comparing with a Gaussian one, and thus it provides a potential model for applications. We also investigate the usual stochastic ordering, the convex ordering, and the peakedness ordering of K-combined random fields and of multivariate K-combined distributions, with necessary and/or sufficient conditions derived.

Correction to: Geometric Properties of the Products of Modified Bessel Functions of the First Kind

Correction to: Geometric Properties of the Products of Modified Bessel Functions of the First Kind

Quasi-Rational Analytic Approximation for the Modified Bessel Function I1(x) with High Accuracy

A new simple and accurate expression to approximate the modified Bessel function of the first kind I1(x) is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA. This method uses the power series of the Bessel function, its asymptotic expansion, and a process of optimization to fit the parameters of a fitting function. The fitting expression is formed by elementary functions combined with rational ones. In the present work, a sum of hyperbolic functions was selected as elementary functions to capture the first two terms of the asymptotic expansion of I1(x), which represents an important improvement with respect to previous research, where just the leading term of the asymptotic series was captured. The new approximation function presents a remarkable agreement with the analytical solution I1(x), decreasing the maximum relative error in more than one order of magnitude with respect to previous similar expressions. Concretely, the relative error was reduced from 10−2 to 4×10−4, opening the possibility of applying the new improved method to other Bessel functions. It is also remarkable that the new approximation is valid for all positive and negative values of the argument.

$${\mathbb {Q}}$$-linear dependence of certain Bessel moments

Let \(I_0\) and \(K_0\) be modified Bessel functions of the zeroth order. We use Vanhove’s differential operators for Feynman integrals to derive upper bounds for dimensions of the \({\mathbb {Q}}\)-vector space spanned by certain sequences of Bessel moments $$\begin{aligned} \left\{ \left. \int _0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm {d}t\right| k\in {\mathbb {Z}}_{\ge 0}\right\} , \end{aligned}$$where a and b are fixed non-negative integers. For \(a\in {\mathbb {Z}}\cap [1,b)\), our upper bound for the \( {\mathbb {Q}}\)-linear dimension is \(\lfloor (a+b-1)/2\rfloor \), which improves the Borwein–Salvy bound \(\lfloor (a+b+1)/2\rfloor \). Our new upper bound \(\lfloor (a+b-1)/2\rfloor \) is not sharp for \(a=2, b=6\), due to an exceptional \({\mathbb {Q}}\)-linear relation \(\int _0^\infty [I_0(t)]^2[K_0(t)]^6 t\mathrm {d}t=72\int _0^\infty [I_0(t)]^2[K_0(t)]^6 t^{3}\mathrm {d}t\), which is provable by integrating modular forms.

Bounds for an integral of the modified Bessel function of the first kind and expressions involving it

Simple upper and lower bounds are obtained for the integral ∫0xe−γttνIν(t)dt, x>0, ν>−12, 0<γ<1. Most of our bounds for this integral are tight as x→∞. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.

Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes

The bulk scaling limit of eigenvalue distribution on the complex plane of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the D-dimensional complex spaces , , in which the Ginibre DPP is realized when D = 1. This one-parameter family () of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each D, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius R in . We prove that any DPP in the Heisenberg family is in the hyperuniform state of class I, in the sense that the number variance behaves as R 2D−1 as R → ∞. Our exact results provide asymptotic expansions of the number variances in large R.

A sequence of Appell polynomials and the associated Jakimovski–Leviatan operators

We are concerned with a special sequence of Appell polynomials, related to the Renyi and Tsallis entropies for the binomial distribution. The generating function is investigated: it is logarithmically convex and has remarkable connections with the modified Bessel function $$I_0(t)$$ and with the index of coincidence for Poisson distribution. The specific form of the Appell polynomials leads to specific properties of the associated Jakimovski–Leviatan operators.

Indefinite integrals involving the exponential integral function

The exponential integral function Ei(x) is given as an indefinite integral of an elementary expression. This allows a second-order linear differential equation for the function to be constructed, which is of conventional form. A limitless number of differential equations can be derived from the original by elementary transformations, and many integrals are given by applying the method of fragments to some of these transformed equations. Results are presented here both for simple transformations and other transformations obtained by solving simple Riccati equations. Some of the Integrals are presented combine Ei(x) with Bessel functions, modified Bessel functions and Whittaker functions. All results have been checked by differentiation using Mathematica.

On two extensions of the canonical Feller\u2013Spitzer distribution

We introduce two extensions of the canonical Feller–Spitzer distribution from the class of Bessel densities, which comprise two distinct stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions with monotone densities, whose upper tails exhibit a power decay. The densities of the members of the first class are expressed in terms of the modified Bessel function of the first kind, whereas the members of the second class have the densities of their Lévy measure given by virtue of the same function. The Laplace transforms for both these families possess closed–form representations in terms of specific hypergeometric functions. We obtain the explicit expressions by virtue of the particular parameter value for the moments of the distributions considered and establish the monotonicity of the mean, variance, skewness and excess kurtosis within the families. We derive numerous properties of members of these classes by employing both new and previously known properties of the special functions involved and determine the variance function for the natural exponential family generated by a member of the second class.

Exponential spectro-temporal modulation generation.

Traditionally, real-time generation of spectro-temporally modulated noise has been performed on a linear amplitude scale, partially due to computational constraints. Experiments often require modulation that is sinusoidal on a logarithmic amplitude scale as a result of the many perceptual and physiological measures which scale linearly with exponential changes in the signal magnitude. A method is presented for computing exponential spectro-temporal modulation, showing that it can be expressed analytically as a sum over linearly offset sidebands with component amplitudes equal to the values of the modified Bessel function of the first kind. This approach greatly improves the efficiency and precision of stimulus generation over current methods, facilitating real-time generation for a broad range of carrier and envelope signals.

EHD instability of two rigid rotating dielectric columns in porous media

The present work analyses the linear azimuthal stability of a single interface between two cylindrical dielectric fluids. The theoretical model consists of two incompressible rotating electrified fluids throughout the porous media. The system is influenced by a uniform azimuthal electric field. The inner cylinder is filled with a viscous liquid. The outer one is occupied by an inviscid gas. The problem meets its motivation from a geophysics point of view. Therefore, for more convenience, the problem is considered in a planar configuration. Typically, the normal mode analysis is used to facilitate the stability approach. The examination resulted in a stream function, which is governed by a fourth-order ordinary differential equation with complicated variable coefficients. By means of the Mathematica software along with the special functions, the distribution of the stream function is written in terms of the modified Bessel functions. A non-dimensional procedure exposes some non-dimensional numbers, for instance, Weber, Ohnesorg, Taylor, Rossby and Darcy numbers. These numbers are considered with regard to the temporal and spatial increase of both frequency and modulation. The linear stability theory generated a very complicated transcendental dispersion equation. The influences of various physical parameters in the stability profile were studied as well.

Multi-parameter Mathieu, and alternating Mathieu series

The main purpose of this paper is to present a multi–parameter study of the familiar Mathieu series and the alternating Mathieu series S(r) and S˜(r). The computable series expansions of the their related integral representations are obtained in terms of higher transcendental hypergeometric functions like Lauricella’s hypergeometric function FC(m)[x], Fox–Wright Ψ function, Srivastava–Daoust S generalized Lauricella function, Riemann Zeta and Dirichlet Eta functions, while the extensions concern products of Bessel and modified Bessel functions of the first kind, hyper–Bessel and Bessel–Clifford functions. Auxiliary Laplace–Mellin transforms, bounding inequalities for the hyper–Bessel and Bessel–Clifford functions are established- which are also of independent but considerable interest. A set of bounding inequalities are presented either for the hyper–Bessel and Bessel–Clifford functions which are to our best knowledge new, or also for all considered extended Mathieu–type series. Next, functional bounding inequalities, log–convexity properties and Turán inequality results are presented for the investigated extensions of multi–parameter Mathieu–type series. We end the exposition by a thorough discussion closes the exposition including important details, bridges to occuring new questions like the similar kind multi–parameter treatment of the complete Butzer–Flocke–Hauss Ω function which is intimately connected with the Mathieu series family.