Numerical Methods for Special Functions

Abstract

Special functions arise in many problems of pure and applied mathematics, statistics, physics, and engineering. This book provides an up-to-date overview of methods for computing special functions and discusses when to use them in standard parameter domains, as well as in large and complex domains. The first part of the book covers convergent and divergent series, Chebyshev expansions, numerical quadrature, and recurrence relations. Its focus is on the computation of special functions. Pseudoalgorithms are given to help students write their own algorithms. In addition to these basic tools, the authors discuss methods for computing zeros of special functions, uniform asymptotic expansions, Pad approximations, and sequence transformations. The book also provides specific algorithms for computing several special functions (Airy functions and parabolic cylinder functions, among others). Audience: This book is intended for researchers in applied mathematics, scientific computing, physics, engineering, statistics, and other scientific disciplines in which special functions are used as computational tools. Some chapters can be used in general numerical analysis courses. Contents: List of Algorithms; Preface; Chapter 1: Introduction; Part I: Basic Methods. Chapter 2: Convergent and Divergent Series; Chapter 3: Chebyshev Expansions; Chapter 4: Linear Recurrence Relations and Associated Continued Fractions; Chapter 5: Quadrature Methods; Part II: Further Tools and Methods. Chapter 6: Numerical Aspects of Continued Fractions; Chapter 7: Computation of the Zeros of Special Functions; Chapter 8: Uniform Asymptotic Expansions; Chapter 9: Other Methods; Part III: Related Topics and Examples. Chapter 10: Inversion of Cumulative Distribution Functions; Chapter 11: Further Examples; Part IV: Software. Chapter 12: Associated Algorithms; Bibliography; Index.