Related papers: A Recursive Method for Computing Certain Bessel Fu…
We introduce a version of the asymptotic expansions for Bessel functions $J_\nu(z)$, $Y_\nu(z)$ that is valid whenever $|z| > \nu$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the…
We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the…
The Bessel-Neumann expansion (of integer order) of a function $g:\mathbb{C}\rightarrow\mathbb{C}$ corresponds to representing $g$ as a linear combination of basis functions $\phi_0,\phi_1,\ldots$, i.e., $g(z)=\sum_{\ell = 0}^\infty w_\ell…
We consider the asymptotic expansion of the Mathieu-Bessel series \[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed,…
We consider the asymptotic expansion of the Mathieu-Bessel series \[S_{\nu,\gamma}^{\mu}(a,b)=\sum_{n=1}^\infty \frac{n^\gamma K_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu>0, \nu\geq 0, b>0, \gamma\in {\bf R})\] as $|a|\to\infty$ in…
A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schl\"{o}milch and Fourier--Bessel expansions in $\mathcal{O}(N(\log N)^2/\log\!\log N)$ operations. The…
In this paper we study the following Bessel series $\sum _{l=1}^{\infty } {J_{l+m'}(r)J_{l+m}(r)}{(l+\beta)^\alpha}$ for any $m,m'\in\mathbb{Z}$, $\alpha\in\mathbb{R}$ and $\beta>-1$. They are a particular case of the second type Neumann…
This paper presents the equality of finite index sums of Bessel func- tions containing arbitrary numbers of terms. These reduce to the familiar three term recursion formulas in simple cases.
This work is an extension of previous work by Alazah et al. [M. Alazah, S. N. Chandler-Wilde, and S. La Porte, Numerische Mathematik, 128(4):635-661, 2014]. We split the computation of the Fresnel Integrals into 3 cases: a truncated Taylor…
We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form \[\sum_{n=1}^\infty e^{-an^p}\frac{J_\nu(an^px)}{(an^px/2)^\nu}\qquad (x>0),\] in the limit $a\to0$, where…
In this work, series expansions in terms of Bessel functions of the first kind are given for the sine and cosine integrals. These representations differ from many of the known Neumann-type series expansions for the sine and cosine…
We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique we propose is based on the formal reduction of these family of functions to Gaussians.
We extend the recursion formula for matrix Bessel functions, which we obtained previously, to superspace. It is sufficient to do this for the unitary orthosymplectic supergroup. By direct computations, we show that fairly explicit results…
This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identites for these functions. The integrals are often used in…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the…
We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of…
A four-term recurrence relation for squared spherical Bessel functions is shown to yield closed-form expressions for several types of finite weighted sums of these functions. The resulting sum rules, which may contain an arbitrarily large…
Through the theory of Jack polynomials we give an iterative method for integral formula of Dunkl-Bessel functions of type $A_{N-1}$ and a partial product formula for it.
We introduce a new representation for the rescaled Appell polynomials and use it to obtain asymptotic expansions to arbitrary order. This representation consists of a finite sum and an integral over a universal contour (i.e. independent of…