Engineering Approach to Asymptotics and Approximations

The objective of this textbook is twofold—to introduce advanced novel mathematical techniques and then to use them to solve a variety of problems that may arise in various applications. The novel mathematical techniques, covered in Part I of the text, involve the use of generalized hypergeometric functions, including the Meijer G-function. We first present an introduction to asymptotic analysis for both small and large arguments of special functions and then use these asymptotic expressions to develop accurate algebraic approximations for generalized hypergeometric functions. In Part II we apply these novel mathematical techniques to a number of problems in electromagnetic wave propagation, imaging through atmospheric turbulence, and non-Kolmogorov turbulence. The idea is to present the mathematical details that are often missing in the literature in the development of various statistical models in these application areas and to introduce new methods for solution of the challenging equations that result from optical turbulence. In many cases we present exact results followed by accurate algebraic approximations. All required mathematical identities and integrals are presented in five appendices at the end of the text. In writing this text, the authors have tried to present the material in sufficient detail to reach an audience with limited knowledge of the subjects covered. To aid the reader in this regard, the text comprises a large number of worked examples in most chapters.