Introduction to Asymptotics and Special Functions

Introduction to Asymptotics and Special Functions is a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable are discussed, along with contour integrals and differential equations with regular and irregular singularities. The Liouville-Green approximation is also considered. Comprised of seven chapters, this volume begins with an overview of the basic concepts and definitions of asymptotic analysis and special functions, followed by a discussion on integrals of a real variable. Contour integrals are then examined, paying particular attention to Laplace integrals with a complex parameter and Bessel functions of large argument and order. Subsequent chapters focus on differential equations having regular and irregular singularities, with emphasis on Legendre functions as well as Bessel and confluent hypergeometric functions. A chapter devoted to the Liouville-Green approximation tackles asymptotic properties with respect to parameters and to the independent variable, eigenvalue problems, and theorems on singular integral equations. This monograph is intended for students needing only an introductory course to asymptotics and special functions. Content: Front Matter, Page iii Copyright, Page iv PREFACE, Page ix PREFACE TO ASYMPTOTICS AND SPECIAL FUNCTIONS, Pages xi-xii 1 - INTRODUCTION TO ASYMPTOTIC ANALYSIS, Pages 1-30 2 - INTRODUCTION TO SPECIAL FUNCTIONS, Pages 31-65 3 - INTEGRALS OF A REAL VARIABLE, Pages 66-105 4 - CONTOUR INTEGRALS, Pages 106-138 5 - DIFFERENTIAL EQUATIONS WITH REGULAR SINGULARITIES; HYPERGEOMETRIC AND LEGENDRE FUNCTIONS, Pages 139-189 6 - THE LIOUVILLE-GREEN APPROXIMATION, Pages 190-228 7 - DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULARITIES; BESSEL AND CONFLUENT HYPERGEOMETRIC FUNCTIONS, Pages 229-278 ANSWERS TO EXERCISES, Pages 279A-280A REFERENCES, Pages 281A-287A INDEX OF SYMBOLS, Pages 289A-290A GENRAL INDEX, Pages 291A-297A