Global asymptotics for multiple integrals with boundaries.
type: Item
instance of: scholarly article
title: Global asymptotics for multiple integrals with boundaries.
zbMATH Open document ID: 1060.30049
DOI: 10.1215/S0012-9074-02-11221-6
author: C. J. Howls
author: E. Delabaere
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The authors extend results given by \textit{C. J. Howls} [Proc. R. Soc. Lond., Ser. A 453, No. 1966, 2271--2294 (1997; Zbl 1067.58501)] for integrals of the following type: \[ I_\Gamma(k)= \int_\Gamma e^{-kf(z)}g(z)\,dz^{(1)}\wedge \cdots\wedge dz^{(n)}, \] where \(f,g:{\mathbb C}\to{\mathbb C}\) are analytic functions in \(z=(z^{(1)},\dots,z^{(n)})\) and \(\Gamma\) is a chain of real dimension \(n\) with \(m\leq n\) smooth irreducible affine hypersurfaces as boundaries. Based on a geometric decomposition of the chain \(\Gamma\) [cf. \textit{F. Pham}, Astérisque 130, 11--47 (1985; Zbl 0597.32012)] the asymptotic expansion follows from a local analysis near the critical points. In a well accessible manner the authors study the asymptotic expansion from a geometrical and an analytical point of view. They analyse Stokes phenomenon (resp. Picard-Lefschetz formulae) and calculate exact remainder terms for truncated asymptotic expansions. Further on, Stokes constants and hyperasymptotics are given. An example illustrates the derived techniques.
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 32C99
Mathematics Subject Classification ID: 34M35
zbMATH DE Number: 1820889
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: Riemann-Debye saddle point method
zbMATH Keywords: Lefschetz thimble
zbMATH Keywords: hyperasymptotics
zbMATH Keywords: Stokes phenomenon
zbMATH Keywords: resurgence properties
zbMATH Keywords: Stoke's constants
reviewed by: Wolfgang zu Castell
MaRDI profile type: Publication
cites work: Q4402443
cites work: Q3217281
cites work: Solution of the Schrödinger equation in terms of classical paths
cites work: Hyperasymptotics for integrals with saddles
cites work: Algorithme de calcul du polynome de Bernstein. Cas non dégéneré. (Algorithm of computation of Bernstein polynomials. The non-degenerate case)
cites work: Monodromy of isolated singularities of hypersurfaces
cites work: Q3041331
cites work: Milnor numbers and the topology of polynomial hypersurfaces
cites work: Q4893922
cites work: Les courants residuels associes à une forme méromorphe
cites work: Q3474020
cites work: Q4288564
cites work: Hyperasymptotics for multidimensional Laplace integrals with boundaries
cites work: Q4261277
cites work: Q3764252
cites work: Singularities and topology of hypersurfaces
cites work: Q4765553
cites work: Five Definitions of Critical Point at Infinity
cites work: Q3737734
cites work: Q3986535
cites work: Q4039743
cites work: Q5515607
cites work: Q3825320
cites work: Q4712071
cites work: Q4866832
cites work: Q4735398
cites work: Families of polynomials with total Milnor number constant
cites work: Hyperasymptotics for integrals with finite endpoints
cites work: Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem
cites work: Q5515310
cites work: Q5843703
cites work: Exponentially improved stationary phase approximations for double integrals
cites work: Asymptotics of a class of multidimensional Laplace–type integrals. I. Double integrals
cites work: Asymptotics of a class of multidimensional Laplace–type integrals. II. Treble integrals
cites work: Le calcul différentiel et intégral sur une variété analytique complexe. (Problème de Cauchy. III.)
cites work: Asymptotic expansions of Laplace-type integrals. III
cites work: Formal solutions of linear meromorphic differential systems and summability
cites work: Intégrales asymptotiques et monodromie
cites work: Multidimensional stationary phase approximation: Boundary stationary point
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: On the bifurcation set of a polynomial function and Newton boundary
cites work: Q4953992
cites work: Hyperdeterminants. I
cites work: Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one
cites work: Hyperterminants. II
cites work: Q4059363
cites work: Q4160104
cites work: Q4855214
cites work: Q5539559
cites work: Q3668374
cites work: Q4204848
cites work: Huygens principle and complex trajectories or Balian and Bloch twenty years after
cites work: Singularities at infinity and their vanishing cycles
cites work: Q3933097
cites work: Asymptotic exponential integrals. Newton's diagram, and the classification of minimal points

Global asymptotics for multiple integrals with boundaries. (cites work)
cites work: Q4323302

Global asymptotics for multiple integrals with boundaries. (cites work)
cites work: Stratifications de Whitney et théorème de Bertini-Sard

Global asymptotics for multiple integrals with boundaries. (cites work)
cites work: Q5343650

Global asymptotics for multiple integrals with boundaries. (cites work)
cites work: Q3253193

Global asymptotics for multiple integrals with boundaries. (cites work)
cites work: Q3993278

Global asymptotics for multiple integrals with boundaries. (full work available at URL)
full work available at URL: s0012-9074-02-11221-6

Global asymptotics for multiple integrals with boundaries. (OpenAlex ID)
OpenAlex ID: W2066698580