ÈŶ¯·¨

Preface1 Algebraic equations1.1 Iteration and expansionlterative methodExpansion method1.2 Singular perturbations and rescalingIterative methodExpansion methodRescaling in the expansion method1.3 Non-integral powersFinding the ezpansion sequencelterative method1.4 Logarithms1.5 Convergence1.6 Eigenvalue problemsSecond order perturbationsMultiple rootsDegenerate roots2 Asymptotic approximations2.1 Convergence and asymptoticness2.2 Definitions2.3 Uniqueness and manipulations2.4 Why asymptotic?Numerical use of diverging series2.5 Parametric expansions2.6 Stokes phenomenon in the complex plane3 Integrals3.1 Watson's lemmaApplication and explanation3.2 Integration by parts3.3 Steepest descentsGlobal considerationsLocal considerationsExample: Stirling's formulaExample: Airy function3.4 Non-local contributionsExample IExample 2Splitting a range of integrationLogarithms3.5 An integral equation: the electrical capacity of a long slender body4 Regular perturbation problems in partial differential equations4.1 Potential outside a near sphere4.2 Deformation of a slowly rotating self-gravitating liquid mass4.3 Nearly'uniform inertial flow past a cylinder5 Matched asymptotic expansion5.1 A linear problem5.1.1 The exact solution5.1.2 The outer approximation5.1.3 The inner approximation (or boundary layer solution)5.1.4 Matching5.1.5 Van Dyke's matching rule5.1.6 Choice of stretching5.1.7 Where is the boundary layer?5.1.8 Composite approximations5.2 Logarithms5.2.1 The problem and initial observations5.2.2 Approximation for r fixed as e\05.2.3 Approximation for p = er fixed as e \05,2.4 Matching by intermediate variable5.2.5 Further terms5.2.6 Failure of Van Dyke's matching rule