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334 results

Page 1 of 17

  • 5 - Multivariate Fourier–Laplace integrals

  • from Part II - Mathematical Background
  • Robin Pemantle, University of Pennsylvania, Mark C. Wilson, University of Massachusetts, Amherst, Stephen Melczer, University of Waterloo, Ontario
  • Book:
    Analytic Combinatorics in Several Variables
    Published online:
    08 February 2024
    Print publication:
    15 February 2024, pp 114-133

  • Summary

    This chapter develops methods to compute asymptotics of multivariate Fourier–Laplace integrals in order to derive general saddle point approximations for use in later chapters. Our approach uses contour deformation, differing from common treatments relying on integration by parts: this requires analyticity rather than just smoothness but is better suited to integration over complex manifolds.

  • 7 - Overview of analytic methods for multivariate GFs

  • from Part III - Multivariate Enumeration
  • Robin Pemantle, University of Pennsylvania, Mark C. Wilson, University of Massachusetts, Amherst, Stephen Melczer, University of Waterloo, Ontario
  • Book:
    Analytic Combinatorics in Several Variables
    Published online:
    08 February 2024
    Print publication:
    15 February 2024, pp 169-220

  • Summary

    This chapter gives a high-level overview of analytic combinatorics in several variables. Stratified Morse theory reduces the derivation of coefficient asymptotics for a multivariate generating function to the study of asymptotic expansions of local integrals near certain critical points on the generating function’s singular set. Determining exactly which critical points contribute to asymptotic behavior is a key step in the analysis . The asymptotic behavior of each local integral depends on the local geometry of the singular variety, with three special cases treated in later chapters.

  • 1 - Introduction

  • from Part I - Combinatorial Enumeration
  • Robin Pemantle, University of Pennsylvania, Mark C. Wilson, University of Massachusetts, Amherst, Stephen Melczer, University of Waterloo, Ontario
  • Book:
    Analytic Combinatorics in Several Variables
    Published online:
    08 February 2024
    Print publication:
    15 February 2024, pp 3-16

  • Summary

    This first chapter motivates our detailed study of the behavior of multivariate sequences, and overviews the techniques we derive using the Cauchy Integral Formula, residues, topological arguments, and asymptotic approximations. Basic asymptotic notation and concepts are introduced, including the background necessary to discuss multivariate expansions.

Page 1 of 17