Preface 1 Algebraic equations 1.1 Iteration and expansion lterative method Expansion method 1.2 Singular perturbations and rescaling Iterative method Expansion method Rescaling in the expansion method 1.3 Non-integral powers Finding the ezpansion sequence lterative method 1.4 Logarithms 1.5 Convergence 1.6 Eigenvalue problems Second order perturbations Multiple roots Degenerate roots 2 Asymptotic approximations 2.1 Convergence and asymptoticness 2.2 Definitions 2.3 Uniqueness and manipulations 2.4 Why asymptotic? Numerical use of diverging series 2.5 Parametric expansions 2.6 Stokes phenomenon in the complex plane 3 Integrals 3.1 Watson's lemma Application and explanation 3.2 Integration by parts 3.3 Steepest descents Global considerations Local considerations Example: Stirling's formula Example: Airy function 3.4 Non-local contributions Example I Example 2 Splitting a range of integration Logarithms 3.5 An integral equation: the electrical capacity of a long slender body 4 Regular perturbation problems in partial differential equations 4.1 Potential outside a near sphere 4.2 Deformation of a slowly rotating self-gravitating liquid mass 4.3 Nearly'uniform inertial flow past a cylinder 5 Matched asymptotic expansion 5.1 A linear problem 5.1.1 The exact solution 5.1.2 The outer approximation 5.1.3 The inner approximation (or boundary layer solution) 5.1.4 Matching 5.1.5 Van Dyke's matching rule 5.1.6 Choice of stretching 5.1.7 Where is the boundary layer? 5.1.8 Composite approximations 5.2 Logarithms 5.2.1 The problem and initial observations 5.2.2 Approximation for r fixed as e\0 5.2.3 Approximation for p = er fixed as e \0 5,2.4 Matching by intermediate variable 5.2.5 Further terms 5.2.6 Failure of Van Dyke's matching rule …… 6 Method of strained co-ordinates 7 Method of multiple scales 8 Improved convergence Bibliography Index
