Square Of Bessel Function Research Articles

Some integrals involving squares of Bessel functions and generalized Legendre polynomials

This paper develops new integral formulas intended for detailed studies of electromagnetics normal modes in spherical and spherical annular cavities.

Surface Relief Modulated Grating in Azo Polymer-From the Tailoring of Diffraction Order to Reshaping of a Laser Beam.

Among light-responsive materials for photonics, azo polymers occupy an important position due to their optical response and the related concepts of consecutive applications. However, scientific insight is still needed to understand the effects of irradiation on the modification of the azo polymer structure and the effect of this modification on incoming probing light. In this work, we report on a surface relief grating with a maximum depth of a record-high value of 1.7 µm, inscribed holographically in a custom synthesized glassy azo polymer belonging to the poly(ether imide) family. We show that the specifically deformed polymer, forming an amplitude-modulated relief grating, has a unique dual effect on an incoming light beam of different diameters. When illuminated by a narrow probe beam, the structure acts as a variable-depth grating, enabling a continuous tuning of the diffraction efficiencies in the entire theoretically predicted range and, thus, generating or eliminating diffracted waves of specified order. Alternatively, when illuminated by a wide probe beam, the whole structure acts as an optical component reshaping the Gaussian light intensity profile into the profiles resembling the squares of Bessel functions of the zeroth- or higher orders. Moreover, a physical justification of the effects observed is provided.

Discrete index transformations with squares of Bessel functions

Discrete analogues of the index transforms with squares of Bessel functions of the first and second kind are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

Integral Characteristics of Wave Packets in the Problem of the Evolution of A Wave Function on A One-Dimensional Lattice

We consider the quantum dynamics of charge transfer on a lattice in the tight-binding approximation and analytically calculate the integral characteristics of the wave packet propagating along the lattice. We focus on calculating the mean and root-mean-square displacements. We also obtain expressions for higher-order moments as series for squares of Bessel functions, which might be independently interesting.

Index transforms with the squares of Bessel functions

ABSTRACTNew index transforms, involving the squares of Bessel functions of the first kind as the kernel, are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third-order partial differential equation, involving the Laplacian, is obtained.

Analysis of multi-frequency subspace migration weighted by natural logarithmic function for fast imaging of two-dimensional thin, arc-like electromagnetic inhomogeneities

The present study seeks to investigate mathematical structures of a multi-frequency subspace migration weighted by the natural logarithmic function for imaging of thin electromagnetic inhomogeneities from measured far-field pattern. To this end, we designed the algorithm and calculated the indefinite integration of square of Bessel function of order zero of the first kind multiplied by the natural logarithmic function. This is needed for mathematical analysis of improved algorithm to demonstrate the reason why proposed multi-frequency subspace migration contributes to yielding better imaging performance, compared to previously suggested subspace migration algorithm. This analysis is based on the fact that the singular vectors of the collected Multi-Static Response (MSR) matrix whose elements are the measured far-field pattern can be represented as an asymptotic expansion formula in the presence of such inhomogeneities. To support the main research results, several numerical experiments with noisy data are illustrated.

Exact formula for the sum of the squares of the Bessel function and the Neumann function of the same order of half-odd integer

Sum of the squares of the Bessel function and the Neumann function of the same order of half-odd integer has been found to be very useful in addressing a puzzle in nuclear physics. One approximate formula available in the literature is valid for the complex argument whose real part is greater than zero, and the absolute value of error term is undefined for half-odd integers. Another approximate formula which is valid for all complex arguments has been obtained using sophisticated mathematical method called Barnes' method. However, the error in the formula is very difficult to calculate. We have obtained exact formula for the sum of the squares of Bessel and Neumann functions of the same order of half-odd integers which is valid for all complex arguments, and its proof is also given.Keywords: Bessel functions; Neumann functions; Anomalous absorption; Partial waves DOI: 10.4038/josuk.v4i0.2694J Sci.Univ.Kelaniya 4 (2008): 15-20

Partial fractions expansions and identities for products of Bessel functions

We derive several partial fractions expansions for products of Bessel functions, and use them to prove algebraic relationships between infinite series involving squares of Bessel functions. We also give formulas for the Taylor series coefficients of the zeros of Bessel functions, when the zeros are regarded as functions of the order x of Jx(y)⁠.

On the index integral transformation with Nicholson's function as the kernel

The integral transformation, which is associated with the Nicholson function as the kernel, is introduced and investigated in the paper. This transformation is an integral, where integration is with respect to an index of the sum of squares of Bessel functions of the first and second kind. Composition representations and relationships with the Meijer K-transform, the Kontorovich–Lebedev transform, the Mellin transform, and the sine Fourier transform are given. We also present boundedness properties, a Parseval type equality, and an inversion formula.

Remarks on the derivation of sum rules of special functions

In this paper we exploit the generating-function method to deduce sum rules involving squares of Bessel functions and Laguerre polynomials.

Some series of squares of Bessel functions

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.