G. Maino

Theory of Generalized Bessel Functions. - II.

SummaryIn this paper we continue the systematic study of the generalized Bessel functions (GBF) recently introduced and often encountered in problems of scattering for which the dipole approximation is inadequate. We analyse the relations among different GBF and discuss their importance for the solution of differential finite-difference equation of the Raman-Nath type. We present numerical results for the first-kind cylinder GBF in the preasymptotic region and also a preliminary analysis of the asymptotic properties of the modified GBF.

Generating functions of multivariable generalized Bessel functions and Jacobi‐elliptic functions

It is pointed out that the Jacobi‐elliptic functions are the natural basis to get generating functions of the multivariable generalized Bessel functions. Analytical and numerical results are given of interest for applications.

Theory of two‐index Bessel functions and applications to physical problems

In this article the theory of two‐index Bessel functions is presented. Their generating function, series expansion, and integral representations are discussed. Their usefulness in physical problems is also discussed in the context of analysis of radiation emitted by relativistic electrons in two‐frequency undulators. Finally, the theoretical analysis proving addition and multiplication theorems for two‐index Bessel functions are completed and their modified forms are introduced.

Phase‐space dynamics and Hermite polynomials of two variables and two indices

The theory of Hermite polynomials of two variables and two indices is discussed herein. Within the context of phase‐space formulation of classical and quantum mechanics, they play the same role as conventional Hermite polynomials in ordinary quantum mechanics. Finally their extension to m variables and m indices is analyzed.

Analytical treatment of the high-gain free electron laser equation

Exact solutions are obtained for the monodimensional and higher dimensional Free Electron Laser high-gain equations. These equations, which belong to the class of integrodifferential Volterra equations are treated within the context of a perturbative approach, yielding suitable closed-form expressions for the relevant solutions. The proposed method allows to obtain a unified analytical formalism for the full FEL dynamics. Numerical results are given for the one-dimensional case where a suitable expression has been derived for the solution in terms of easily computable functions.

A numerical method for generalized exponential integrals

Abstract We present a method for evaluation of the exponential integral, E s ( x ), generalized to an arbitrary order s > 0. The algorithm is valid whatever s > 0 and x > 0. In the region x ⩾ 1, we start from a proper initial value, obtained by asymptotic calculation, and then compute the required E s ( x ) by means of a suitable combination of Taylors expansions and recurrences, whatever s > 0. When x s 0 ( x ) (0 s 0 ⩽ 1), which is obtained by the means of suitable expansions. A forward recursion finally yields the required E s ( x ). Numerical stability and accuracy of the proposed algorithm are discussed and some results given.

A note on the theory of n -variable generalized bessel functions

SummaryIn this note we introduce a further generalization of Bessel-type functions, discussing the case of a multivariables and one-index function. This kind of function can be usefully exploited in problems in which the dipole approximation does not hold and many higher harmonics are simultaneously operating. We analyse the relevant recurrence properties, the modified forms and the generating functions.

Recent results for generalized exponential integrals

Basic properties of the exponential-integral function of real order, Ev(x), and relevant expressions for evaluating this special function are presented. The mathematical results have been essentially obtained by generalizing known formulae valid for the usual exponential-integral, En(x).

Analytical and numerical results on M -variable generalized Bessel functions

Recently, some multivariable special functions have been obtained by generalizing functions of Bessel type. Here, we continue the treatment of these functions starting fromJn(x, y; i), which is of noticeable practical interest. Finally, we consider the cases of functionsJn(x1, x2,..., xM) and the related modified version,In(x1, x2,..., xM), with two significant physical applications. Calculations of multivariable generalized Bessel functions are discussed and numerical results are given forJn(x1,x2;i), withn=0, 1, in a region of interest.

Theory of generalized Hermite polynomials

Abstract We introduce multivariable generalized forms of Hermite polynomials and analyze both the Gould-Hopper type polynomials and more general forms, which are analogues of the classical orthogonal polynomials, since they represent a basis in L 2( R N) Hilbert space, suitable for series expansion of square summable functions of N variables: Moreover, the role played by these generalized Hermite polynomials in the solution of evolution-type differential equations is investigated: The key-note of the method leading to the multivariable polynomials is the introduction of particular generating functions, following the same criteria underlying the theory of multivariable generalized Bessel functions.