From Circular to Bessel Functions: A Transition through the Umbral Method

The critical length of a space of functions can be described as the supremum of the length of the intervals where Hermite interpolation problems are unisolvent for any choice of nodes. We analyze the critical length for spaces containing products of algebraic polynomials and trigonometric functions. We show the relation of these spaces with spherical Bessel functions and bound above their critical length by the first positive zero of a Bessel function of the first kind.

Partial wave expansion is encountered in a wide range of problems in theoretical physics. Typically, spherical Bessel functions are encountered in scattering of nuclei, atoms, and molecules. A routine is described for computing spherical Bessel and Neumann functions jl( x) and nl( x), respectively, by backward and forward recurrence relations. The method is fast, very accurate, not code intensive, and easily adopted for a personal computer. These features make the code well suited for the physics student research environment.

Summary Biological tissues are complex system that contains a variety of liquid components, macromolecules and ions. They are highly heterogeneous media that consist of various compartments and barriers of different diffusivities. In terms of its cyto-histologic architecture, a tissue can be regarded as a porous structure made up of a set of more or less connected compartments in a network like arrangement. Since molecular motion of water is significantly affected by macromolecules, the variation in the relaxation times between tissues is attributed to the effect of macromolecular interaction. The movement of water molecules during diffusion-driven random displacement is restricted by compartmental boundaries and other molecular obstacle in such a way that the actual diffusion distance is reduced, compare with what is expected in unrestricted diffusion. In this study, based on the new NMR diffusion equation, Bessel functions and the spherical harmonic functions, we present an analysis of diffusing blood component by monitoring its NMR signal emitted at its unique resonant frequency. This view can be very important in the study of pathologies such as sickle cell anemia and finding a treatment to the genetic disorder. This is because, we may be able to visualize blood vessels pores that are blocked by the sickle cell and find out the factor that makes the erythrocytes rigid. It is expected that the diamagnetic property of the oxyhemoglobin that makes it difficult to deliver the oxygen on it would be very useful here. Hence, a very slowly decaying signal or non - decaying signal could be the indication of some form of problem with the vessel. Background Analytical solutions to the Bloch NMR flow equations can provide a straightforward means of obtaining fundamental and accurate information on the translational motion of nuclear spins for dynamic analysis of fluid flow in biological and non biological systems. However, the interpretation of the data is complicated by the effects of restricting geometries. The mathematics required to model diffusion coefficient for diffusion within restricting geometries especially for medical imaging has been very difficult because analytical solutions to the Bloch NMR flow equations are generally considered not possible. Therefore, numerical and approximate solutions are the only options available for the analysis of special arrangement of porous media such as biological and liquid crystals systems. In this contribution, we have presented the analytical solutions to the inhomogeneous second order differential NMR diffusion equation in cylindrical and spherical coordinates for theoretical, computational and possible experimental studies of fluids in restricted geometries. Methods

A finite integral involving the product of powers of trigonometric functions, up to two associated Legendre functions, and zero or one Bessel function is evaluated. When certain combinations of the otherwise complex function parameters are integers, the resulting expression becomes greatly simplified. So restricting the parameters, this still quite general case may be transformed into four canonical forms, each of which admits rapid convergence of the only nonterminating series in the expressions. Finally, closed form expressions are obtained for a number of special cases.

An oscillating infinite series involving product of Bessel function J o(x) and an oscillating infinite series involving trigonometric function sin(x) were evaluated and computed numerically in [1] and [2] respectively. In this paper, an oscillating infinite series involving product of exponential, Bessel and trigonometric functions is evaluated. The series is transformed first into the sum of two infinite integrals by using contour integration and then the infinite integral with oscillating integrand is transformed through some identities into a finite integral containing modified Bessel function K 1(x). Finally, theset two integrals are evaluated numerically without any computational difficulties at all.

The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.

This paper explores a novel application of the Laplace transform in cryptographic schemes by employing a linear combination of Bessel functions and exponential functions. By leveraging the transform’s properties, we develop an encryption-decryption framework that is computationally efficient and highly secure against certain types of cryptographic attacks. The exponential decay observed in Laplace-transformed domains and the complex behaviour of Bessel functions are powerful tools for data encoding and concealment. Analytical and numerical results validate the method’s effectiveness, showcasing its potential for modern cryptographic systems. Cryptography has long been a cornerstone of secure communication, with its methods evolving in tandem with technological advancements. While traditional schemes rely on number theory or algebraic structures, this paper explores an alternative approach that combines Laplace transforms with special functions. Specifically, we focus on employing Bessel functions in conjunction with exponential functions to construct an encryption algorithm. The Laplace transform’s unique properties, such as linearity, time-shifting, and frequency domain representation, offer a fertile ground for developing innovative cryptographic methods. Then the algorithm described in this study generates a dynamic cipher by processing plaintext using polynomial, exponential, and factorial-based operations. The four fundamental steps comprising the computational process are text preprocessing, dynamic coefficient calculation, encryption packaging, and reverse analysis. The computational cost of each step varies depending on the number of characters processed. The symbol for this character quantity is n. The cryptographic analysis of the problem is discussed in detail.

In this appendix we summarize the properties of Bessel functions, spherical Bessel functions and related functions. These functions are required in the solution of the radial Schrodinger equation and in the derivation of formulae for scattering amplitudes and cross sections. They are also important in the discussion of other topics in this monograph including multiphoton processes. For more complete presentations of their properties, reference should be made to specialized monographs on mathematical functions such as Abramowitz and Stegun [1], Watson [956] and Whittaker and Watson [964].

When deriving dyadic Green's functions for the spherical structures with uniaxial or gyrotropic anisotropic materials and studying the interaction among the neutrons, ions and atoms, some semi-infinite integrals whose integrant function consists of two spherical Bessel functions and a power function needs to be evaluated. Furthermore, those kinds of integrals are of great importance to the research of atomic particle collisions, multipole moments and relativistic processes. During the recent work associated with the formulations, we evaluated in detail the integrals of two spherical Bessel functions given in C.T. Whelan (1986), but found the mistakes in the solution given in C.T. Whelan (1986). Therefore, this paper revisited the evaluation of the integral and provided the correct solution to the integral in spherical coordinates in terms of distribution, step functions and delta functions. The formulation was further extended and it is also found that the solution varies differently in the cases of even and odd values of l - l'.

The conference 'From DNA-Inspired Physics to Physics-Inspired Biology' (1–5 June 2009, International Center for Theoretical Physics, Trieste, Italy) that myself and two former presidents of the American Biophysical Society—Wilma Olson (Rutgers University) and Adrian Parsegian (NIH), with the support of an ICTP team (Ralf Gebauer (Local Organizer) and Doreen Sauleek (Conference Secretary)), have organized was intended to establish stronger links between the biology and physics communities on the DNA front. The relationships between them were never easy. In 1997, Adrian published a paper in Physics Today ('Harness the Hubris') summarizing his thoughts about the main obstacles for a successful collaboration. The bottom line of that article was that physicists must seriously learn biology before exploring it and even having an interpreter, a friend or co-worker, who will be cooperating with you and translating the problems of biology into a physical language, may not be enough. He started his story with a joke about a physicist asking a biologist: 'I want to study the brain. Tell me something about it!' Biologist: 'First, the brain consists of two parts, and..' Physicist: 'Stop. You have told me too much.'

In circular tomosyntesis obtained from a sufficient number of radiographs with projection directions forming a single cone, object detail at a distance from the plane of interest is blurred according to the zero order Bessel function. The main lobe of this window function defines slice thickness, while its ringing side lobes are undesirable because they permit further outlying structures to “leak” through. Using the orthogonality of Bessel functions, a sampling scheme consistmg of multiple, concentric projection cones was designed to synthesize by a finite FourierBessel series a slice window with superior side lobe suppression. The window that concentrates the most “energy” within a finite slice, and can be realized by a limited number of sampling cones, is the zero order circular prolate spheroidal function. Its application to tomosynthesis of 3 mm thick slices with a characteristic detail size of 0.5 mm, concentrated 95% of the total series expansion to only 3 concentric circular scans, yielding a theoretical suppression of the first side lobe of − 38 dB, to be compared to − 8 dB achieved with the Bessel function. Experimental implementation of this optimal sampling scheme using 3, 8, and 14 projection views, distributed over cones with opening half‐anges of 1.92, 4.39, and 6.88°, respectively, resulted in a side‐lobe attenuation of at least − 20 dB. This attenuation reduced significantly artifacts arising from out‐of‐plane detail of high spatial frequency (e.g., edges) in tomosynthetic images of clinical interest in dentistry.

When deriving dyadic Green's functions for the spherical structures with gyrotropic or bianisotropic materials, an integral whose integrand function consists of two spherical Bessel functions and a power function needs to be evaluated. Therefore, this paper revisits thoroughly the evaluation of the integral of Il,l'(kappa,kappa'). Starting from pointing out an error, it provides the correct solution to the integral in spherical coordinates in terms of distribution, in particular, step functions and delta functions. The formulation is further extended to a more generalized integral Hl,l'lambda(kappa,kappa'); and it is newly found that the solution to the generalized integral varies differently in the cases of even and odd values of l-l'. The mistakes that we found in the previous literature can also be proved easily by some of our intermediate solutions

There are many applications for problems involving thermal conduction in two-dimensional cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green’s functions where the principle of superposition is employed in combining a one-dimensional cylindrical case with a one-dimensional Cartesian case to provide a temperature solution for a two-dimensional cylindrical. Green’s functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

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.

Supply chain stability is a main feature of effective operation. In order to analyze the stability problem of supply chain system under the shock load, Bessel function has been used to simulate the stress wave propagation in supply chain system under the external shock load. The momentum properties of system's wave function are given by the Fourier transform, and it has been shown by superposition of supply chain inherent wave and shock load transfer wave. The supply chain stability conditions have been given by linear continuous system stable condition and discrete dynamic system stable condition. The stability critical state of supply chain system had been proposed by effective management span value through system's simulation analysis.