Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Asymptotic Generalities
- 2 Matched Asymptotic Expansions
- 3 The WKB Method
- 4 The Method of Multiple Scales
- 5 Approximations of Integrals
- 6 Discrete Asymptotics
- 7 Divergent Series and Exponentially Small Phenomena
- 8 Solutions to Exercises
- Appendix Inhomogeneous Linear ODEs
- References
- Index
1 - Asymptotic Generalities
Published online by Cambridge University Press: 28 October 2026
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Asymptotic Generalities
- 2 Matched Asymptotic Expansions
- 3 The WKB Method
- 4 The Method of Multiple Scales
- 5 Approximations of Integrals
- 6 Discrete Asymptotics
- 7 Divergent Series and Exponentially Small Phenomena
- 8 Solutions to Exercises
- Appendix Inhomogeneous Linear ODEs
- References
- Index
Summary
This chapter introduces the main concepts required to discuss the methods presented in the rest of the book. Using simple algebraic examples containing a small parameter, the crucial difference between regularly and singularly perturbed problems is introduced. It is explained in which sense asymptotic expansions can approximate solutions of equations. Importantly, it is discussed what manipulation can or cannot always be done on these expansions. While regular perturbations are generally amenable to straightforward resolution, this is not so with singularly perturbed problems. The latter almost inevitably leads to the coexistence of slow and fast scales in the solutions being studied, the treatment of which requires the use of the techniques developed in subsequent chapters.
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- Practical Asymptotics , pp. 1 - 17Publisher: Cambridge University PressPrint publication year: 2026
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