A convergent and asymptotic Laplace method for integrals
Fecha
2023Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión publicada / Argitaratu den bertsioa
Impacto
|
10.1016/j.cam.2022.114897
Resumen
Watson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F (z) := ∫ ∞
0
e
−zf (t)
g(t)dt for large values of the parameter z. They are useful
tools in the asymptotic approximation of special functions that have a Laplace integral representation. But in most of the important examples of special functions, the
asymptotic expansion derived by means of Watson’s lem ...
[++]
Watson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F (z) := ∫ ∞
0
e
−zf (t)
g(t)dt for large values of the parameter z. They are useful
tools in the asymptotic approximation of special functions that have a Laplace integral representation. But in most of the important examples of special functions, the
asymptotic expansion derived by means of Watson’s lemma or Laplace’s method is not
convergent. A modification of Watson’s lemma was introduced in [Nielsen, 1906] where,
by the use of inverse factorial series, a new asymptotic as well as convergent expansion
of F (z), for the particular case f (t) = t, was derived. In this paper we go some steps
further and investigate a modification of the Laplace’s method for F (z), with a general
phase function f (t), to derive asymptotic expansions of F (z) that are also convergent,
accompanied by error bounds. An analysis of the remainder of this new expansion shows
that it is convergent under a mild condition for the functions f (t) and g(t), namely, these
functions must be analytic in certain starlike complex regions that contain the positive
axis [0,∞). In many practical situations (in many examples of special functions), the
singularities of f (t) and g(t) are off this region and then this method provides asymptotic
expansions that are also convergent. We illustrate this modification of the Laplace’s
method with the parabolic cylinder function U(a, z), providing an asymptotic expansions
of this function for large z that is also convergent. [--]
Materias
Asymptotic expansions of integrals,
Watson’s lemma,
Laplace’s method,
Convergent expansions,
Special functions
Editor
Elsevier
Publicado en
Journal of Computational and Applied Mathematics 422 (2023) 114897
Departamento
Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas /
Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika Saila /
Universidad Pública de Navarra/Nafarroako Unibertsitate Publikoa. Institute for Advanced Materials and Mathematics - INAMAT2
Versión del editor
Entidades Financiadoras
This research was supported by the Universidad Pública de Navarra, grant PRO-UPNA (6158) 01/01/2022. Open access
funding provided by Universidad Pública de Navarra.