Related papers: Asymptotic behavior of zeros of Bessel function de…

It is our aim to establish a general analytic theory of asymptotic expansions of type f(x)=a_1 phi_1(x)+dots+ a_n phi_n(x)+o(phi_n(x)), x tends to x_0 (*), where the given ordered n-tuple of real-valued functions phi_1 dots,phi_n forms an…

We produce an estimate for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin \theta$ for a fixed parameter $0\leq \theta\leq \pi/2$ or $t= y…

We derive the exact asymptotics of \[ P\left( \sup_{t\ge 0} \Bigl( X_1(t) - \mu_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - \mu_2 s\Bigr)> u \right), \ \ u\to\infty, \] where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional…

Similar to the associated Legendre functions, the differential equation for the associated Bessel functions $B_{l,m}(x)$ is introduced so that its form remains invariant under the transformation $l\rightarrow -l-1$. A Rodrigues formula for…

We examine the asymptotic behaviour of the zeros of sections of the binomial expansion. That is, we consider the distribution of zeros of $\displaystyle B_{r,n}(z) = \sum_{k=0}^r {n \choose k} z^k$, where $1 \le r < n$.

Asymptotic expansions for generalised trigonometric integrals are obtained in terms of elementary functions, which are valid for large values of the parameter $a$ and unbounded complex values of the argument. These follow from new…

In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of…

We prove an asymptotic for the sum of $\zeta^{(n)} (\rho)X^{\rho}$ where $\zeta^{(n)} (s)$ denotes the $n$th derivative of the Riemann zeta function, $X$ is a positive real and $\rho$ denotes a non-trivial zero of the Riemann zeta function.…

The asymptotic behavior of the convolution-integral of a special form of the Airy function and a function of the power-like behavior at infinity is obtained. The integral under consideration is the solution of the Cauchy problem for an…

One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le…

We study asymptotical expansion as $\nu\to0$ for integrals over ${ \mathbb{R} }^{2d}=\{(x,y)\}$ of quotients of the form $F(x,y) \cos(\lambda x\cdot y) \big/ \big( (x\cdot y)^2+\nu^2\big)$, where $\lambda\ge 0$ and $F$ decays at infinity…

We propose an asymptotic expansion formula for matrix integrals, including oscillatory terms (derivatives of theta-functions) to all orders. This formula is heuristically derived from the analogy between matrix integrals, and formal matrix…

We present a rational extension of Newton diagram for the positivity of ${}_1F_2$ generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots $\beta(\alpha)$ of \begin{align*}…

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the…

The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of…

This paper deals with the study of the zeros of the big $q$-Bessel functions. In particular, we prove a new orthogonality relations for this functions similar to the one for the classical Bessel functions. Also we give some applications…

Uniform asymptotic expansions are derived for Whittaker's confluent hypergeometric functions $M_{\kappa,\mu}(z)$ and $W_{\kappa,\mu}(z)$, as well as the numerically satisfactory companion function $W_{-\kappa,\mu}(ze^{-\pi i})$. The…

The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function…

Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and…

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in [11]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential…