# Showing papers on "Bessel function published in 2002"

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TL;DR: GTOBAS as mentioned in this paper is a program for fitting Gaussian-type orbitals (GTOs) to Bessel and Coulomb functions over a finite range, where the exponents of the GTOs are optimized using the method of Nestmann and Peyerimhoff [J. Phys. B 23 (1990) L773].

Abstract: GTOBAS is a program for fitting Gaussian-type orbitals (GTOs) to Bessel and Coulomb functions over a finite range. The exponents of the GTOs are optimized using the method of Nestmann and Peyerimhoff [J. Phys. B 23 (1990) L773]. The appended module NUMCBAS provides the numerical Bessel and Coulomb functions required as input for the program. The use of GTO continuum basis sets is particularly important in electron–molecule scattering calculations when polyatomic targets are involved. Sample results for such calculations are also discussed.

188 citations

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25 Jul 2002

TL;DR: The theory of Bessel functions and its application in the theory of Oscillations, Hydrodynamics, and Heat Transfer were discussed in this paper, with a focus on Bessel Equation.

Abstract: Foundation of the Theory of Bessel Functions Bessel Equation. Properties of Bessel Functions. Definite and Improper Integrals. Series in Bessel Functions. Applications of Bessel Functions. Problems of the Theory of Plates and Shells. Problems of the Theory of Oscillations, Hydrodynamics and Heat Transfer. Appendix A. Brief Information on Gamma Functions.

168 citations

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TL;DR: Using L/sup 2/-metric on the set of Bessel K forms, a pseudometric on the image space for quantifying image similarities/differences is proposed and some applications, including clutter classification and pruning of hypotheses for target recognition, are presented.

Abstract: Seeking probability models for images, we employ a spectral approach where the images are decomposed using bandpass filters and probability models are imposed on the filter outputs (also called spectral components). We employ a (two-parameter) family of probability densities, called Bessel K forms, for modeling the marginal densities of the spectral components, and demonstrate their fit to the observed histograms for video, infrared, and range images. Motivated by object-based models for image analysis, a relationship between the Bessel parameters and the imaged objects is established. Using L/sup 2/-metric on the set of Bessel K forms, we propose a pseudometric on the image space for quantifying image similarities/differences. Some applications, including clutter classification and pruning of hypotheses for target recognition, are presented.

115 citations

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TL;DR: In this article, a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel, was shown to be positivity-preserving.

Abstract: It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for non-negative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial hereby means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.

110 citations

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TL;DR: New bounds are proposed for the Marcum Q-function, which is defined by an integral expression where the 0th-order modified Bessel function appears, and prove to be very tight and outperform bounds previously proposed in the literature.

Abstract: New bounds are proposed for the Marcum Q-function, which is defined by an integral expression where the 0th-order modified Bessel function appears. The proposed bounds are derived by suitable approximations of the 0th-order modified Bessel function in the integration region of the Marcum Q-function. They prove to be very tight and outperform bounds previously proposed in the literature. In particular, the proposed bounds are noticeably good for large values of the parameters of the Marcum Q-function, where previously introduced bounds fail and where exact computation of the function becomes critical due to numerical problems.

91 citations

01 Jan 2002

TL;DR: In this paper, the authors present a finite-difference equation for waveguide eigenmode solution valid for a dielectric corner, which is based on an expansion of the field components as powers of the radius at the corner.

Abstract: We present a discussion of the behavior of the electric and magnetic fields satisfying the two-dimensional Helmholtz equation for waveguides in the vicinity of a dielectric corner. Although certain components of the electric field have long been known to be infinite at the corner, it is shown that all components of the magnetic field are finite, and that finite-difference equations may be derived for these fields that satisfy correct boundary conditions at the corner. These finite-difference equations have been combined with those derived in the previous paper to form a full-vector waveguide solution algorithm of unprecedented accuracy. This algorithm is employed to provide highly accurate solutions for the fundamental modes of a previously studied standard rib waveguide structure. Indeed, with second-order accurate code it may not be worth the trouble to change the way corners are handled. However, for those intent on more accurate simulations, correct treatment of the corner points is essential. The history of the behavior of fields near a dielectric or metal wedge dates to the early work of Bouwcamp and Meixner (2), (3), who pointed out some of the peculiarities of this problem, including field components that diverge weakly at the corner as a small negative power of the distance from the corner (ra- dius). Meixner further published a solution formalism (3) based on an expansion of the field components as powers of the ra- dius. This expansion was later shown to be incorrect by An- derson and Solodukhov (4), and although papers continued to appear, almost ten years elapsed before a correct expansion was published. Makarov and Osipov (5) explained the reason for the problems with Meixner's series and correctly identified the missing terms as containing various powers of the logarithm of the radius. However, the latter authors did not treat a "gen- eral" corner, but considered only a single field component, and left their expansion in a very general form quite unsuitable for the derivation of finite-difference equations. This entire body of work, and in fact the existence of the problem itself, re- mained largely unknown to the optics community until 1992, when Sudbo (6) described the importance of the problem in the modeling of optical waveguide structures. Apparently unaware of Makarov and Osipov's work, he correctly pointed out the in- herent difficulties of including corner effects in any numerical algorithm for modeling waveguides. Since that time only a few authors have attempted to include corner effects in their mod- eling work (7), (8) and no one has attempted a thorough deriva- tion. This paper derives a finite-difference equation for waveguide eigenmode solution valid for a dielectric corner. This equation is ostensibly first-order accurate, and when combined with highly accurate difference equations in the interior regions and at simple planar boundaries (1), has resulted in a highly accurate full-vector waveguide eigenmode solver whose trun- cation error ranges from second to third order. This departure from the expected third-order behavior apparently results from the presence of infinite derivatives at the corner and will be discussed in more depth below. The present work may be viewed as an extension of the work of Makarov and Osipov (5), in that the standard solutions of the Helmholtz equation in a uniform region (i.e., Bessel functions and sines and cosines) are augmented by terms containing logarithms that appear similar to their solutions. However, here we consider the full vector

91 citations

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TL;DR: A generalization of a representation theorem due to Caratheodory is used to derive new families of Gaussian-type quadratures for weighted integrals of exponential functions and their applications to integration and interpolation of bandlimited functions.

Abstract: We introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions. We use a generalization of a representation theorem due to Caratheodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. For a given accuracy ϵ, selecting an eigenvalue close to ϵ yields an approximate quadrature with that accuracy. To compute its weights and nodes, we present a new fast algorithm. These new quadratures can be used to approximate and integrate bandlimited functions, such as prolate spheroidal wave functions, and essentially bandlimited functions, such as Bessel functions. We also develop, for a given precision, an interpolating basis for bandlimited functions on an interval.

87 citations

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TL;DR: In this article, a methodology is presented that allows the derivation of low-truncation-error finite-difference representations or the two-dimensional Helmholtz equation, specific to waveguide analysis.

Abstract: A methodology is presented that allows the derivation of low-truncation-error finite-difference representations or the two-dimensional Helmholtz equation, specific to waveguide analysis. This methodology is derived from the formal infinite series solution involving Bessel functions and sines and cosines. The resulting finite-difference equations are valid everywhere except at dielectric corners, and are highly accurate (from fourth to sixth order, depending on the type of grid employed). None the less, they utilize only a nine-point stencil, and thus lead to only minor increases in numerical effort compared with the standard Crank-Nicolson equations.

82 citations

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TL;DR: In this paper, the authors considered the problem of a point force acting in an unbounded, three-dimensional, isotropic elastic solid, where the material is inhomogeneous; it is "functionally graded" and the Lame moduli vary exponentially in a given fixed direction.

Abstract: The problem of a point force acting in an unbounded, three–dimensional, isotropic elastic solid is considered. Kelvin solved this problem for homogeneous materials. Here, the material is inhomogeneous; it is ‘functionally graded’. Specifically, the solid is ‘exponentially graded’, which means that the Lame moduli vary exponentially in a given fixed direction. The solution for the Green9s function is obtained by Fourier transforms, and consists of a singular part, given by the Kelvin solution, plus a non–singular remainder. This grading term is not obtained in simple closed form, but as the sum of single integrals over finite intervals of modified Bessel functions, and double integrals over finite regions of elementary functions. Knowledge of this new fundamental solution for graded materials permits the development of boundary–integral methods for these technologically important inhomogeneous solids.

80 citations

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TL;DR: A simple correspondence between the paraxial propagation formulas along the optical axis of a uniaxial crystal and inside an isotropic medium is found in the case of beams with linearly polarized circularly symmetric boundary distributions.

Abstract: A simple correspondence between the paraxial propagation formulas along the optical axis of a uniaxial crystal and inside an isotropic medium is found in the case of beams with linearly polarized circularly symmetric boundary distributions. The electric fields of the ordinary and the extraordinary beams are related to the corresponding expressions in a medium with refractive index no and ne2/no, where no and ne are the ordinary and the extraordinary refractive indexes, respectively. Closed-form expressions for Laguerre–Gauss and Bessel–Gauss beams propagating through an anisotropic crystal are given.

73 citations

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TL;DR: Low-divergence, high-brightness harmonic emission has been generated by using a fundamental beam with a truncated Bessel intensity profile using the hollow-fiber compression technique, which allows one to optimize both temporal and spatial characteristics of the high-order harmonic generation process.

Abstract: Low-divergence, high-brightness harmonic emission has been generated by using a fundamental beam with a truncated Bessel intensity profile. Such a beam is directly obtained by using the hollow-fiber compression technique, which indeed allows one to optimize both temporal and spatial characteristics of the high-order harmonic generation process. This is particularly important for the applications of radiation, where extreme temporal resolution and high brightness are required.

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TL;DR: In this paper, an exact analysis of the free vibrations of a simply supported, homogeneous, transversely isotropic, cylindrical panel is presented in the context of L ord-Shulman (L S), Green-L indsay (GL), and Green-Nagdhi (GN) theories of thermoelasticity.

Abstract: In this article, based on three-dimensional thermoelasticity, an exact analysis of the free vibrations of a simply supported, homogeneous, transversely isotropic, cylindrical panel is presented in the context of L ord-Shulman (L S), Green-L indsay (GL), and Green-Nagdhi (GN) theories of thermoelasticity. Three displacement potential functions are introduced so that the equations of motion and heat conduction are uncoupled and simplified. It is noticed that the purely transverse mode is independent of temperature change and rest of the motion. The equations for free vibration problems are further reduced to four second-order ordinary differential equations after expanding the potential and temperature functions with an orthogonal series. A modified Bessel function solution with complex arguments is then directly used for complex eigenvalues. Numerical examples are presented to clarify the developed method and compare the results to the existing one.

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TL;DR: In this paper, the variation of the vorticity of the superposition of two Bessel singular beams under free-space propagation is analyzed, and it is shown that in the near field the combined beam creates light pattern with much richer vortex content than that of individual beams.

Abstract: The variation of the vorticity of the superposition of two Bessel singular beams under free-space propagation is analyzed. We demonstrate that in the near field the combined beam creates light pattern with much richer vortex content than that of individual beams. Under diffraction the combined beam dynamically evolves into the beam with rather simple vortical structure in the far field. A qualitative agreement of experimental results with theoretical predictions is obtained.

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TL;DR: In this article, a computational method is presented to investigate SH waves in functionally graded material (FGM) plates, in which the material properties are assumed as a quadratic function in the thickness direction, and a general solution for the equation of motion governing the QLE has been derived.

Abstract: A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.

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TL;DR: The objective is first to establish, using only the q -Jackson integral and the q-derivative, some properties of this function with proofs similar to the classical case; second to construct the associated q-Fourier analysis which will be used in a coming work to constructThe q-analogue of the Bessel-hypergroup.

Abstract: In this paper we study the q-analogue of the j"@a Bessel function (see (1)) which results after minor changes from the so-called Exton function studied by Koornwinder and Swarttow. Our objective is first to establish, using only the q -Jackson integral and the q-derivative, some properties of this function with proofs similar to the classical case; second to construct the associated q-Fourier analysis which will be used in a coming work to construct the q-analogue of the Bessel-hypergroup.

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TL;DR: In this article, an exact random variate generation for the Bessel distribution is discussed. But the expected time of the algorithm is uniformly bounded over all choices of the parameters, and the algorithm avoids any computation of Bessel functions or Bessel ratios.

Abstract: In this paper, we discuss efficient exact random variate generation for the Bessel distribution. The expected time of the algorithm is uniformly bounded over all choices of the parameters, and the algorithm avoids any computation of Bessel functions or Bessel ratios.

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TL;DR: In this article, simple closed-form expressions for partition functions associated with relativistic bosons and fermions in odd spatial dimensions were developed, including the effects of a nontrivial Polyakov loop and generalizing well-known high temperature expansions.

Abstract: We develop exact, simple closed form expressions for partition functions associated with relativistic bosons and fermions in odd spatial dimensions. These expressions, valid at high temperature, include the effects of a nontrivial Polyakov loop and generalize well-known high temperature expansions. The key technical point is the proof of a set of Bessel function identities which resum low temperature expansions into high temperature expansions. The complete expressions for these partition functions can be used to obtain one-loop finite temperature contributions to effective potentials, and thus free energies and pressures.

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TL;DR: In this article, the Bessel process with time-varying dimension is generalized to the extended Cox-Ingersoll-Ross model with time varying parameters, and a special class of extended CIR models is studied.

Abstract: We study the Bessel processes withtime-varying dimension and their applications to the extended Cox-Ingersoll-Rossmodel with time-varying parameters. It is known that the classical CIR model is amodified Bessel process with deterministic time and scale change. We show thatthis relation can be generalized for the extended CIR model with time-varyingparameters, if we consider Bessel process with time-varying dimension. Thisenables us to evaluate the arbitrage free prices of discounted bonds and theircontingent claims applying the basic properties of Bessel processes. Furthermorewe study a special class of extended CIR models which not only enables us to fitevery arbitrage free initial term structure, but also to give the extended CIRcall option pricing formula.

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TL;DR: In this paper, the problem of infinite-barrier lens-shaped quantum dots (QDs) in parabolic rotational coordinates was solved using the Frobenius method and also by first transforming the separated differential equations into the Whittaker equation.

Abstract: We solve the problem of infinite-barrier lens-shaped quantum dots (QDs) in parabolic rotational coordinates exactly. The solutions are obtained directly using the Frobenius method and also by first transforming the separated differential equations into the Whittaker equation. We have obtained a new relation connecting the Bessel wavefunctions to the Whittaker functions. Results are given for both symmetrical and asymmetrical QDs. Studies of the energy spectra at constant volume were also performed; it is found that the shape dependence is very different from those found in previous studies of QDs with ellipsoidal and elliptic shapes.

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TL;DR: In this article, an efficient approach for improving convergence of semi-infinite very oscillatory integrals, based on the nonlinear D-transformation and some useful properties of spherical Bessel, reduced Bessel and sine functions, is presented.

Abstract: Two-electron, four-center Coulomb integrals are undoubtedly the most difficult type involved in ab initio and density functional theory molecular structure calculations. Millions of such integrals are required for molecules of interest; therefore rapidity is the primordial criterion when the precision has been reached. This work presents an extremely efficient approach for improving convergence of semi-infinite very oscillatory integrals, based on the nonlinear D-transformation and some useful properties of spherical Bessel, reduced Bessel, and sine functions. The new method is now shown to be applicable to evaluating the two-electron, four-center Coulomb integrals over B functions. The section with numerical results illustrates the unprecedented efficiency of the new approach in evaluating the integrals of interest.

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TL;DR: In this paper, the generalized pencil of function method was used to evaluate the Green's functions of single-layer and multilayer structures, and closed-form expressions for the method-of-moments matrix coefficients were obtained.

Abstract: This paper presents an efficient technique to evaluate the Green's functions of single-layer and multilayer structures. Using the generalized pencil of function method, a Green's function in the spectral domain is accurately approximated by a short series of exponentials, which represent images in spatial domain. New compact closed-form spatial-domain Green's functions are found from these images using several semi-infinite integrals of Bessel functions. With the numerical integration of the Sommerfeld integrals avoided, this method has the advantages of speed and simplicity over numerical techniques, and it leads to closed-form expressions for the method-of-moments matrix coefficients. Numerical examples are given and compared with those from numerical integration.

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TL;DR: In this paper, two uniform asymptotic representations of the Bessel function Jα(x) and the Airy function Ai(x), respectively, are presented for the Laguerre polynomial Ln α(x).

Abstract: Some of the work on the construction of inequalities and asymptotic approximations for the zeros λn,k(α), k = 1,2 .... ,n, of the Laguerre polynomial Lnα(x) as v = 4n + 2α + 2 → ∞, is reviewed and discussed. The cases when one or both parameters n and α unrestrictedly diverge are considered. Two new uniform asymptotic representations are presented: the first involves the positive zeros of the Bessel function Jα(x), and the second is in terms of the zeros of the Airy function Ai(x). They hold for k= 1,2 .... , [qn] and for k = [pn], [pn] + 1 ..... n, respectively, where p and q are fixed numbers in the interval (0, 1 ). Numerical results and comparisons are provided which favorably justify the consideration of the new approximations formulas.

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TL;DR: In this paper, a generalization of type 3 ultrashort pulses (also known as pulse beams or isodiffracting pulses) is introduced, and a model spectral distribution that is zero outside a finite range is investigated.

Abstract: A generalization of type 3 ultrashort pulses (also known as pulse beams or isodiffracting pulses) is introduced. The Bessel beam form of this generalized beam consists of pulses that propagate in free space, without spreading, with a velocity that can be less than that of light. A model spectral distribution that is zero outside a finite range is investigated.

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TL;DR: In this article, it was shown that moments of negative order and positive non-integral order of a nonnegative random variable X can be expressed by the Laplace transform of X.

Abstract: It is shown that moments of negative order as well as positive non-integral order of a nonnegative random variable X can be expressed by the Laplace transform of X Applying these results to certain first passage times gives explicit formulae for moments of suprema of Bessel processes as well as strictly stable Levy processes having no positive jumps

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TL;DR: In this paper, the authors investigated the distribution of nodes at and beyond the classical turning point of idealized chaotic quantum eigenfunctions and derived a formula for the density of nodes in the semiclassical limit.

Abstract: We investigate the distribution of nodes at and beyond the classical turning point of idealized chaotic quantum eigenfunctions. A formula for the density of nodes is derived in the semiclassical limit, and the rate at which this density falls off as one moves into the forbidden region is also studied. The discussion is supported by numerical results. Corrections to the Bessel function correlation in the classically allowed region are necessary for finite and are given here.

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TL;DR: A new family of solutions of the nonparaxial wave equation that represents ultrashort pulsed light beam propagation in free space is found, showing that the even- and odd-order spatially induced dispersions partially compensate for each other to give rise to pulse spreading, weakening, asymmetry, and center shift.

Abstract: We find a new family of solutions of the nonparaxial wave equation that represents ultrashort pulsed light beam propagation in free space. The spatial and temporal parts of these pulsed beams are separable; the spatial transverse part is described by a Bessel function and remains unchanged during propagation, but the temporal profile can be arbitrary. Therefore the pulsed beam exhibits diffraction-free behavior with no transverse spreading, but the temporal part changes as if in a dispensive medium; the change is dominated by what we call spatially induced group-velocity dispersion. The analytical and numerical investigations show that the even- and odd-order spatially induced dispersions partially compensate for each other so as to give rise to pulse spreading, weakening, asymmetry, and center shift.

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TL;DR: In this paper, the authors analyze steady-state heat conduction in a triangular-profile fin using a relatively new, exact series method of solution known as the differential transform method.

Abstract: We analyze steady-state heat conduction in a triangular-profile fin using a relatively new, exact series method of solution known as the differential transform method. This method converges with only six terms or less for the cases considered. Its advantage is that, unlike many popular methods, it is an exact method and yet it does not require the use of Bessel or other special functions

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10 Jul 2002TL;DR: The algorithm is able to find all rational transformations for a large class of functions, in particular the special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions, and can be generalized to equations of higher order.

Abstract: We describe a new algorithm for computing special function solutions of the form y(x) = m(x)F(ξ(x)) of second order linear ordinary differential equations, where m(x) is an arbitrary Liouvillian function, ξ(x) is an arbitrary rational function, and F satisfies a given second order linear ordinary differential equation. Our algorithm, which is based on finding an appropriate point transformation between the equation defining F and the one to solve, is able to find all rational transformations for a large class of functions F, in particular (but not only) the 0F1 and 1F1 special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions. It is also able to identify the values of the parameters entering those special functions, and can be generalized to equations of higher order.

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TL;DR: In this article, the evaluation of the modified Bessel function of the third kind of purely imaginary order Kia(x) is discussed; also, analogous results for the derivative are presented.

Abstract: The evaluation of the modified Bessel function of the third kind of purely imaginary order Kia(x) is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions, and a continued fraction method, depending on the ranges of x and a. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method, and the nonoscillatory integral representation can be used to accurately compute the function Kia(x) in the range 0 ≤ a ≤ 200, 0 ≤ x ≤ 100; using a similar scheme the derivative K'ia(x) can also be computed within these ranges.

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TL;DR: In this paper, an integral representation for the inverse problem is found, from which a formula for initial temperature is derived using Picard's criterion and the singular system of the associated operators.

Abstract: We investigate the inverse problem involving recovery of initial temperature from the information of final temperature profile in a disc. This inverse problem arises when experimental measurements are taken at any given time, and it is desired to calculate the initial profile. We consider the usual heat equation and the hyperbolic heat equation with Bessel operator. An integral representation for the problem is found, from which a formula for initial temperature is derived using Picard's criterion and the singular system of the associated operators.