ASYMPTOTIC ANALYSIS

This volume brings together 30 chapters across five parts, offering a structured, in-depth survey of asymptotic analysis and its applications. Readers will find rigorous treatments of orthogonal polynomials, boundary layer problems, saddle-point integrals, turning point theory, Airy and Bessel expansions, special functions, and modern approaches to difference equations, differential equations, Riemann–Hilbert problems, integrals, and singular perturbation problems. Each chapter blends classical foundations with recent breakthroughs, making the book both a comprehensive reference and a graduate-level textbook with exercises for hands-on learning.

Over the past three decades, asymptotic analysis has become indispensable in number theory, combinatorics, probability and statistics, mathematical physics, engineering, and applied sciences, equipping researchers and students with powerful methods for solving complex problems. This book not only surveys the latest advances in asymptotic methods but also demonstrates their practical applications across diverse mathematical models.

Designed for advanced undergraduate students, graduate students, researchers, and instructors, the text can be used as a reference guide to modern asymptotic techniques or adapted into course modules, strengthening both theoretical understanding and applied problem-solving.

Contents:

• Preface
• About the Authors
• Difference Equations:
  • Asymptotics of Difference Equations with Bounded Coefficients
  • Asymptotics of Difference Equations with General Coefficients
  • Asymptotics of Difference Equations with a Parameter
  • Turning Point Theory: Airy Expansion
  • Turning Point Theory: Bessel Expansion
  • Transition Point at the Origin
  • Transition Point at the Origin Revisited: A Critical Case
• Riemann-Hilbert Problems:
  • Riemann-Hilbert Problem
  • Global Asymptotics of Generalized Freud Orthogonal Polynomials
  • Freud Weight Revisited: New Special Functions
  • Global Asymptotics of the Meixner Polynomials
  • Global Asymptotics of the Stieltjes-Wigert Polynomials
• Differential Equations:
  • Asymptotics of Differential Equations with Irregular Singularities
  • Asymptotics of Differential Equations with a Turning Point
  • Asymptotics of Differential Equations with a Simple Pole
  • Asymptotics of Differential Equations with Two Coalescing Turning Points
• Integrals:
  • Classical Methods
  • Distribution Method
  • Asymptotics of Integrals with a Coalescing Saddle Point
  • Asymptotics of Jacobi Polynomials
  • Asymptotics of a Double Integral
  • Asymptotics of Pollaczek Polynomials
  • Error Bounds for the Steepest Descent Method
  • Error Bounds for the Asymptotic Expansions of the Hermite Polynomials
  • Generalized Bessel Functions
• Singular Perturbations:
  • A Singular Perturbation Problem
  • An Internal Boundary Layer Problem
  • A Boundary-Value Problem with Spurious Solutions
  • A Nested Boundary Layer Problem
  • Number of Solutions to Carrier's Problem
• Bibliography
• Index

Readership: This book is written for advanced undergraduate students, graduate students, postdoctoral researchers, and faculty in applied mathematics, analysis, physics, engineering, and statistics, while also serving as a valuable addition to university libraries supporting advanced coursework and research. It will appeal to interdisciplinary scholars and professionals in computer science, quantitative finance, operations research, and economics who rely on asymptotic and approximation methods for modeling and analysis. Applied scientists and industry researchers in data-intensive and engineering fields will also benefit, and the book doubles as an advanced reference for senior undergraduates preparing for graduate study.