Related papers: Asymptotic expansion of the partition function for…

We demonstrate how the asymptotics for large $|z|$ of the generalised Bessel function \[{}_0\Psi_1(z)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(an+b) n!},\] where $a>-1$ and $b$ is any number (real or complex), may be obtained by exploiting the…

We give an alternative method to that of Hardy-Ramanujan-Rademacher to derive the leading exponential term in the asymptotic approximation to the partition function p(n,a), defined as the number of decompositions of a positive integer 'n'…

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 consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon>1$ and…

We prove asymptotic equivalence of nonparametric additive regression and an appropriate Gaussian white noise experiment in which a multidimensional shifted Wiener process is observed, whose dimension equals the number of additive…

We study the family of Fourier-Laplace transforms $$ F_{\alpha,\beta}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^{\beta}\exp(\mathrm{i} t^{\alpha}-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $\alpha>1$ and…

We investigate the spectral properties of the Dirichlet Laplacian on large finite metric balls within irregular infinite graphs of quadratic volume growth. We consider an exhaustion $G_n = B_{R_n}(x_0)$ and the spectral zeta value $Z_n(1) =…

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[F(a+\epsilon\lambda,m;c+\lambda;x),\qquad \lambda\to+\infty\] for $x<1$ and positive integer $m$ when the parameter $\epsilon>1$ and the constants $a$ and…

We prove asymptotic formulas for the complex coefficients of $(\zeta q;q)_\infty^{-1}$, where $\zeta$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions…

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…

We prove the existence of a 1/N expansion to all orders in beta matrix models with a confining, off-critical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained…

In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a…

A new uniform asymptotic expansion for the incomplete gamma function $\Gamma(a,z)$ valid for large values of $z$ was given by the author in {\it J. Comput. Appl. Math.} {\bf 148} (2002) 323--339. This expansion contains a complementary…

Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree $n$ and parameters $\alpha$ and $\beta$. From these new results, asymptotic expansions of the zeros are derived and methods are…

We investigate an asymptotic expansion of the solution of the master equation under the modulation of control parameters. In this case, the non-decaying part of the solution becomes the dynamical steady state expressed as an infinite series…

In a previous article of the authors with M. Canalis-Durand, monomial asymptotic expansions, Gevrey asymptotic expansions and monomial summability were introduced and applied to certain systems of singularly perturbed differential…

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 examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

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 consider the asymptotic behavior of the incomplete gamma functions gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are needed to describe the transition area z~a in which case error functions are used as main…