Modified Bessel Functions Research Articles

Fast evaluation of the Cooray–Rubinstein formula in time domain via piecewise recursive convolution technique

The Cooray-Rubinstein formula has been widely used for evaluating the horizontal component of the electric field generated by a lightning return stroke over a lossy ground. An accurate and efficient time domain method is presented in this study, by means of piecewise recursive convolution. The recursive convolution is achieved by expressing the kernel function of the ground surface impedance by a piecewise exponential function, in which the integral form of the modified Bessel function of the first kind is evaluated using the trapezoidal numerical integration. Since the time-domain analytical expression of the ground surface impedance kernel function has been adopted, the computational error can be directly controlled within a reasonable tolerance. The validity and efficiency of the proposed method are demonstrated by the extensive comparisons with the method in other literature.

Inequalities for integrals of modified Bessel functions and expressions involving them

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.

Extended Mellin integral representations for the absolute value of the gamma function

Abstract We derive Mellin integral representations in terms of Macdonald functions for the squared absolute value s ↦ | Γ ⁢ ( a + i ⁢ s ) | 2 {s\mapsto|\Gamma(a+is)|^{2}} of the gamma function and its Fourier transform when a < 0 {a<0} is non-integer, generalizing known results in the case a > 0 {a>0} . This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation.

Morse oscillator propagator in the high temperature limit II: Quantum dynamics and spectroscopy

This paper is a continuation of Paper I (Toutounji, 2017) of which motivation was testing the applicability of Morse oscillator propagator whose analytical form was derived by Duru (1983). This is because the Morse oscillator propagator was reported (Duru, 1983) in a triple-integral form of a functional of modified Bessel function of the first kind, which considerably limits its applicability. For this reason, I was prompted to find a regime under which Morse oscillator propagator may be simplified and hence be expressed in a closed-form. This was well accomplished in Paper I. Because Morse oscillator is of central importance and widely used in modelling vibrations, its propagator applicability will be extended to applications in quantum dynamics and spectroscopy as will be reported in this paper using the off-diagonal propagator of Morse oscillator whose analytical form is derived.

A Novel Full-Duplex Primary Signal Extraction Method Based on Polarization Vector Distance in Cognitive Radio Networks

In this paper, a polarization vector distance based signal extraction (PVD-SE) method for full-duplex primary users (FD-PUs) oriented cognitive radio networks (CRN) is studied. The problem of primary signal extraction in FD-PUs oriented CRN boils down to a finite hypotheses testing problem. Based on the orientation and magnitude difference of the signal samples from different FD-PUs, the proposed PVD-SE method can efficiently detect each PU’s signal space by calculating polarization vector distance (PVD) between any two samples. It is found that the samples from the same PU experience relatively smaller PVD in comparison with those from different FD-PUs. As a result of the Gaussian distributed noise, the resulting PVD is non-central chi-squared distributed, which depends on the modified Bessel function of the first kind without closed-form expressions. Using large argument (LA) and small argument closed-form approximations of modified Bessel functions, the optimal extraction threshold that minimizes the extraction error probability is investigated, and analytical expressions of optimal threshold under the LA assumption are derived for amplitude modulated and non-amplitude modulated primary signals. The presented analytical results are verified by Monte Carlo simulation and the performance of PVD-SE is validated.

A Truly Two-Dimensional Discretization of Drift-Diffusion Equations on Cartesian Grids

A genuinely two-dimensional discretization of general drift-diffusion (including incompressible Navier--Stokes) equations is proposed. Its numerical fluxes are derived by computing the radial derivatives of “bubbles” which are deduced from available discrete data by exploiting the stationary Dirichlet--Green function of the convection-diffusion operator. These fluxes are reminiscent of Scharfetter and Gummel's in the sense that they contain modified Bessel functions which allow one to pass smoothly from diffusive to drift-dominating regimes. For certain flows, monotonicity properties are established in the vanishing viscosity limit (``asymptotic monotony'') along with second-order accuracy when the grid is refined. Practical benchmarks are displayed to assess the feasibility of the scheme, including the “western currents” with a Navier--Stokes--Coriolis model of ocean circulation.

Monotonicity and convexity of the ratios of the first kind modified Bessel functions and applications

Monotonicity and convexity of the ratios of the first kind modified Bessel functions and applications

Applying Hankel’s Expansion for Performance Analysis in Double-Nakagami (Generalized-K) Fading Channels

The double-scattering fading models have found applications in many transmission scenarios. In this paper, approximate probability density functions (PDFs) are proposed for double-Nakagami fading and Gamma–Gamma fading, based on Hankel’s expansion of the modified Bessel function of the second kind. The proposed PDF is not only consistent with the exact PDF when the difference between the two shape parameters is equal to an integer plus one half, but also accurately approximates the distribution when the shape parameters are both greater than or equal to 2. The proposed approximate PDF has been compared with the power series expansion and the mixture Gamma model. It is found that the proposed PDF is in simple form and no parameters need to be tabulated. Moreover, the proposed PDF does not require an increasing number of terms to maintain precision with increasing diversity order of the reception scheme. Closed-form outage probability, ergodic capacity, and symbol error rate expressions have been derived using the proposed approximate PDFs for equal-gain combining, selection combining, and two-stage reception schemes. The proposed approximations are also leveraged for performance analysis of space-time block codes and free-space optical links. The precision of the proposed approximations is validated by comparing with the simulation results.

Explicit formula for the density of local times of Markov jump processes

In this note we show a simple formula for the joint density of local times, last exit tree and cycling numbers of continuous-time Markov chains on finite graphs, which involves the modified Bessel function of the first type.

Performance analysis of non-identically distributed FSO systems with dual- and triple-branch based on MRC over Gamma-Gamma fading channels

In this work, the performance of free-space optical (FSO) communication system based on maximal ratio combining using binary phase shift keying subcarrier intensity modulation over Gamma-Gamma fading channels has been studied systematically. Under identically or non-identically distributed branches, the analytical expressions for the bit error rate function of signal-to-noise are derived by expressing the modified Bessel function of second kind with Meijer G-function for dual- and triple-branch systems, respectively. In terms of H-fox function, the new expressions have more general forms and are more efficient for computation. It is found that the dual- and triple-branch systems significantly outperform the direct link system under weak, moderate and strong turbulence conditions. Monte Carlo simulation is also provided to confirm the accuracy of the proposed model.

Monotonicity and convexity of the ratios of the first kind modified Bessel functions and applications

Monotonicity and convexity of the ratios of the first kind modified Bessel functions and applications

Stationary analysis of a fluid queue driven by an M/M/1/N queue with disaster and subsequent repair

This paper studies a fluid queueing model driven by an M/M/1/N queue subject to disaster and subsequent repair. The underlying system of differential difference equations that governs the process are solved using generating function methodology. Explicit expressions for the joint steady state probabilities of the state of the background queueing model and the content of the buffer are obtained in terms of modified Bessel function of the first kind. Numerical illustrations are added to support the theoretical results.

Bivariate Gamma Distribution for Wavelength-Resolution SAR Change Detection

A gamma probability density function (pdf) is shown to be an alternative to model the distribution of the magnitudes of high-resolution, i.e., wavelength-resolution, synthetic aperture radar (SAR) images. As investigated in this paper, it is more appropriate and more realistic statistical in comparison with, e.g., Rayleigh. A bivariate gamma pdf is considered for developing a statistical hypothesis test for wavelength-resolution incoherent SAR change detection. The practical issues in implementation of statistical hypothesis test, such as assumptions on target magnitudes, estimations for scale and shape parameters, and implementation of modified Bessel function, are addressed. This paper also proposes a simple processing scheme for incoherent change detection to validate the proposed statistical hypothesis test. The proposal was experimented with 24 CARABAS data sets. With an average detection probability of 96%, the false alarm rate is only 0.47 per square kilometer.

Extended Riemann-Liouville type fractional derivative operator with applications

Abstract The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.

On a Neumann-type series for modified Bessel functions of the first kind

In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions, while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie et al.’s formula expressing this series as a Φ 2 \Phi _2 -Horn confluent hypergeometric function. In the course of the proof, we shed light on the occurrence of multiple angles in their formula through elementary symmetric functions and get a new representation of Gegenbauer polynomials.

Discrete Bessel functions and partial difference equations

We introduce a new class of discrete Bessel functions and discrete modified Bessel functions of integer order. After obtaining some of their basic properties, we show that these functions lead to fundamental solutions of the discrete wave equation and discrete diffusion equation.

Free energy on a cycle graph and trigonometric deformation of heat kernel traces on odd spheres

We consider a possible ‘deformation’ of the trace of the heat kernel on odd dimensional spheres, motivated by the calculation of the free energy of a scalar field on a discretized circle. By using an expansion in terms of the modified Bessel functions, we obtain the values of the free energies after a suitable regularization.

On complete monotonicity of certain special functions

Given an entire function f ( z ) f(z) that has only negative zeros, we shall prove that all the functions of type f ( m ) ( x ) / f ( n ) ( x ) , m > n f^{(m)}(x)/f^{(n)}(x),\ m>n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q q -Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, q q -exponential functions, q q -Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros

Modified Bessel functions of the first kind Iv (z) (Infeld functions) where v > −1 are considered. Due to the constraint on the parameter v, all zeros of the function Iv (z) except z = 0 are simple, located on the imaginary axis by symmetric pairs and form an infinite countable set. On the basis on previous research of the authors dealing with general Bessel functions of the first kind Jv (z), a question about reciprocal expansion 1/Iv (z) in series of simple fractions of a certain structure (Krein’s series) is studied. The general formulas to calculate of special infinite sums containing degrees of Infeld function zeros are an important consequence of obtained expansion in simple fractions of the value 1/Iv(z) with any v > −1. The possibility of concrete definition of established summation relationships at different values of parameters and their connection with analogous relationships for the Bessel functions of the first kind Jv (z) is discussed.

Analytic approximation for the modified Bessel function I−2/3(x)

In the present work an analytic approximation to modified Bessel function of negative fractional order I−2/3(x) is presented. The validity of the approximation is for every positive value of the independent variable. The accuracy is high in spite of the small number (4) of parameters used. The approximation is a combination of elementary functions with rational ones. Power series and assymptotic expansions are simultaneously used to obtain the approximation.