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Asymptotics and Special Functions
About This Book
Asymptotics and Special Functions provides a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable and contour integrals are discussed, along with the Liouville-Green approximation and connection formulas for solutions of differential equations. Differential equations with regular singularities are also considered, with emphasis on hypergeometric and Legendre functions. Comprised of 14 chapters, this volume begins with an introduction to the basic concepts and definitions of asymptotic analysis and special functions, followed by a discussion on asymptotic theories of definite integrals containing a parameter. Contour integrals as well as integrals of a real variable are described. Subsequent chapters deal with the analytic theory of ordinary differential equations; differential equations with regular and irregular singularities; sums and sequences; and connection formulas for solutions of differential equations. The book concludes with an evaluation of methods used in estimating (as opposed to bounding) errors in asymptotic approximations and expansions. This monograph is intended for graduate mathematicians, physicists, and engineers.
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Table of contents
- Front Cover
- Asymptotics and Special Functions
- Copyright Page
- Table of Contents
- Dedication
- PREFACE
- CHAPTER 1. INTRODUCTION TO ASYMPTOTIC ANALYSIS
- CHAPTER 2. INTRODUCTION TO SPECIAL FUNCTIONS
- CHAPTER 3. INTEGRALS OF A REAL VARIABLE
- CHAPTER 4. CONTOUR INTEGRALS
- CHAPTER 5. DIFFERENTIAL EQUATIONS WITH REGULAR SINGULARITIES; HYPERGEOMETRIC AND LEGENDRE FUNCTIONS
- CHAPTER 6. THE LIOUVILLE-GREEN APPROXIMATION
- CHAPTER 7. DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULARITIES; BESSEL AND CONFLUENT HYPERGEOMETRIC FUNCTIONS
- CHAPTER 8. SUMS AND SEQUENCES
- CHAPTER 9. INTEGRALS: FURTHER METHODS
- CHAPTER 10. DIFFERENTIAL EQUATIONS WITH A PARAMETER: EXPANSIONS IN ELEMENTARY FUNCTIONS
- CHAPTER 11. DIFFERENTIAL EQUATIONS WITH A PARAMETER: TURNING POINTS
- CHAPTER 12. DIFFERENTIAL EQUATIONS WITH A PARAMETER: SIMPLE POLES AND OTHER TRANSITION POINTS
- CHAPTER 13. CONNECTION FORMULAS FOR SOLUTIONS OF DIFFERENTIAL EQUATIONS
- CHAPTER 14. ESTIMATION OF REMAINDER TERMS
- ANSWERS TO EXERCISES
- REFERENCES
- INDEX OF SYMBOLS
- GENERAL INDEX