相关论文: Algorithm xxx: Modified Bessel functions of imagin…

We describe a variety of methods to compute the functions $K_{ia}(x)$, $L_{ia}(x)$ and their derivatives for real $a$ and positive $x$. These functions are numerically satisfactory independent solutions of the differential equation $x^2 w''…

The $\nu$-zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x>0$. In the case of the modified Bessel function of the second kind $K_{i\nu}(x)$, it is known that it possesses a countably infinite…

Bessel functions with pure imaginary index (order) play an important role in corpuscular optics where they govern the dynamics of charged particles in isotrajectory quadrupoles. Recently they were found to be of great importance in…

We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel…

There are three main types of numerical computations for the Bessel function of the second kind: series expansion, continued fraction, and asymptotic expansion. In addition, they are combined in the appropriate domain for each. However,…

The modified Bessel function of the second kind $K_{i\nu}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $\nu_n\sim \pi n/\log\,n$ as…

Bessel and modified Bessel functions of imaginary order $i\nu$ ($\nu >0$) are studied. Asymptotic expansions are derived as $\nu \to \infty$ that are uniformly valid in unbounded complex domains, with error bounds provided. Coupled with…

We examine indefinite integral involving of arbitrary power $x$, multiplied by three spherical Bessel functions of the first kind $j_{h},j_{k}$, and $j_{l}$ with integer order $h,k,l \geq 0$ and an exponential. Then we add some conditions…

The paper presents the derivation of the asymptotic behavior of $\nu$-zeros of the modified Bessel function of imaginary order $K_{{\rm i}\nu}(z)$. This derivation is based on the quasiclassical treatment of the exponential potential on the…

We present an efficient self-contained algorithm for computing the modified Bessel function of the first kind $I_{\nu}(z)$, implemented in a robust Fortran code supporting double and quadruple (quad) precision. The algorithm overcomes the…

To target challenges in differentiable optimization we analyze and propose strategies for derivatives of the Mat\'ern kernel with respect to the smoothness parameter. This problem is of high interest in Gaussian processes modelling due to…

The derivatives with respect to order {\nu} for the Bessel functions of argument x (real or complex) are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series…

We propose an analytical approximation for the modified Bessel function of the second kind $K_\nu$. The approximation is derived from an exponential ansatz imposing global constrains. It yields local and global errors of less than one…

The modified Bessel function of the second kind K$\nu$ appears in a wide variety of applied scientific fields. While its use is greatly facilitated by an implementation in most numerical libraries, overflow issues can be encountered…

A new analytical approximation function is proposed to accurately fit the solution of a fractional differential equation of order one-half, whose nonhomogeneous term is defined by a modified Bessel function of the first kind. The exact…

Let $L f(x):=-\frac{d^2}{dx^2}f(x)-\frac{ r}{x}\frac{d}{dx}f(x),\quad x>0, r>0$ be the Bessel operator on $((0,\infty), |\cdot|, x^rdx)$. In this paper, we prove the sharp weak type $(1,1)$ estimate for the imaginary power $L^{i\alpha},…

In this paper we present two efficient approximations for the complex error function $w \left( {z} \right)$ with small imaginary argument $\operatorname{Im}{\left[ { z } \right]} < < 1$ over the range $0 \le \operatorname{Re}{\left[ { z }…

We determine the asymptotic behaviour of the $n$th derivatives of the Bessel functions $J_\nu(a)$ and $K_\nu(a)$, where $a$ is a fixed positive quantity, as $n\to\infty$. These results are applied to the asymptotic evaluation of two…

Linear second order differential equations of the form $d^{2}w/dz^{2}-\left \{ {u^{2}f\left( u,z\right) +g\left( z\right) }\right\} w=0$ are studied, where $\left| u\right| \rightarrow \infty $ and $z$ lies in a complex bounded or unbounded…

Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to…