An efficient automatic quadrature procedure is developed for numerically computing the integrals 0 , where the function is smooth and nonoscillatory at infinity and is the Bessel functions of order ν =1,0 and 1/4. The procedure involves the use of an automatic integration scheme of modified FFT used for evaluating Fourier integrals and product type integration, and the modified W-transformation...

In the paper, the authors verify the complete monotonicity of the difference e − ψ′(t) on (0,∞), compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of e, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind. These results show us some new properties and relat...

The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermor...

of fundamental Fourier coefficients, which plays an important role in the theory of Bessel models and L-functions. For instance, in certain cases, non-vanishing of a fundamental Fourier coefficient of a cuspidal Siegel modular form F is equivalent to existence of a global Bessel model of fundamental type (cf. [16, Lemma 4.1]) and is used to show analytic properties and special value results for...

for some fixed τ , which is called the period of f . Though function approximation using orthogonal polynomials is very convenient, there is only one kind of periodic polynomial, that is, a constant. So, polynomials are not good for approximating periodic functions. In this case, trigonometric functions are quite useful. A large class of important computational problems falls under the category...

Here are some facts about Fourier Series — useful for pde and elsewhere. Proofs of Lemmas are easy exercises, and not given. On the other hand, proofs of LEMMAS are harder; their proofs are indicated, or a reference is given.

Abstract. The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional ...

We report the first experimental realization of all-optical trapping and manipulation of plasmonic nanowires in three dimensions. The optical beam used for trapping is the Fourier transform of a linearly polarized Bessel beam (termed FT-Bessel). The extended depth of focus of this beam enables the use of a retroreflection geometry to cancel radiation pressure in the beam propagation direction, ...

<p style='text-indent:20px;'>In this paper we establish <i>L<sup>p</sup></i>-boundedness properties for variation operators defined by semigroups associated with Fourier-Bessel expansions.</p>