Bessel Functions and Related Functions

A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed.

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

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.

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.

Analytic solutions of the fundamental equation of the reference spectrum method for nuclear matter are obtained in the cases of the square well and exponential potentials with hard cores. In the former case analytic wavefunctions are obtained for every partial wave L. These wavefunctions are expressed in terms of spherical Bessel and Hankel functions or in terms of spherical Bessel, Hankel, and Newmann functions of order L. In the latter case an analytic expression is obtained for the s wavefunction only. This is given in terms of the transcendental functions G±v(r) and M±v(r) which are defined as series of well-known functions. Finally an application is made of our results to the calculation of the well depth of a Λ particle in its ground state in nuclear matter.

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 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.

Many functions in classical mathematics are largely defined in terms of their derivatives, so Bessel's function is solution of Bessel's equation, etc. For definiteness, we need to add other properties, such as initial values, branch cuts, etc. What actually makes up the definition of a function in computer algebra? The answer turns out to be a combination of arithmetic and analytic properties.

In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations.

Chapter 14 - Series Solutions: Important ODEs

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

CHAPTER 17 - Bessel Functions

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.

In this paper, we have performed experimental, analytical, and numerical studies of beams with topological charges of $\ifmmode\pm\else\textpm\fi{}1$ and $\ifmmode\pm\else\textpm\fi{}2$ formed by silicon binary phase axicons (BPAs) with spiral zone structures. The axicons were illuminated with the Novosibirsk free electron laser radiation (a continuous stream of 100-ps pulses at $f=5.6$ MHz). The cw power of the beams produced reached 30 W and can by doubled via antireflection coating of the axicons. The intensity distribution in the beam cross sections was in good agreement with the Bessel functions and was kept constant within a distance of about $L/r\ensuremath{\approx}190$ and 100, where the first ring radii of the beams r were 0.9 and 1.5 mm for the Bessel beams of the first and second orders, respectively. Although the characteristics of the beams (Bessel cross section, ``diffraction-free'' propagation, self-recovery after passing obstacles, and randomly inhomogeneous media) corresponded to the properties of ideal Bessel beams, their spatial Fourier spectrum (the image in the focal plane of the lens) was, instead of an ideal ring, intertwined segments of arcs with phases shifted by $\ensuremath{\pi}$, the number of which was equal to the double value of the topological charge. This feature can be used, for example, in a demultiplexing unit of a free vortex-wave communication system or for identification of beam topological charge. We also revisited Young's double-slit diffraction and rotation of beams obstructed by a half-plane, previously applied to Laguerre-Gaussian beam characterization, in the case of the Bessel beams. The Young diffraction pattern demonstrated in this case a complicated intensity-phase distribution. It was shown that the Bessel beams formed by BPAs have two important advantages, which can be used in applications, in comparison with other methods of generation, e.g., a combination of an axicon lens with a spiral phase plate. Although the phase jumps of the axicons are designed for a determined wavelength (141 $\ensuremath{\mu}\mathrm{m}$ in our case), the BPAs can form the beams at incident radiation with any wavelength, albeit with a reduced diffraction efficiency, and their cross section is the same for any wavelength.