Various methods for asymptotic evaluation of integrals containing a large parameter, and solutions of ordinary linear differential equations by means of asymptotic expansion.
Various methods for asymptotic evaluation of integrals containing a large parameter, and solutions of ordinary linear differential equations by means of asymptotic expansion.
Introduction; References Chapter I. Asymptotic Series 1.1 O-symbols 1.2 Asymptotic sequences 1.3 Asymptotic expansions 1.4 Linear operations with asymptotic expansions 1.5 Other operations with asymptotic expansions 1.6 Asymptotic power series 1.7 Summation of asymptotic series References Chapter II. Integrals 2.1 Integration by parts 2.2 Laplace integrals 2.3 Critical points 2.4 Laplace's method 2.5 The method of steepest descents 2.6 Airy's integral 2.7 Further examples 2.8 Fourier integrals 2.9 The method of stationary phase References Chapter III. Singularities of Differential Equations 3.1 Classification of singularities 3.2 Normal solutions 3.3 The integral equation and its solution 3.4 Asymptotic expansions of the solutions 3.5 Complex variable. Stokes' phenomenon 3.6 Bessel functions of order zero References Chapter IV. Differential Equations with a Large Parameter 4.1 Liouville's problem 4.2 Formal solutions 4.3 Asymptotic solutions 4.4 Application to Bessel functions 4.5 Transition points 4.6 Airy functions 4.7 Asymptotic solutions valid in the transition region 4.8 Uniform asymptotic representations of Bessel functions References
Introduction; References Chapter I. Asymptotic Series 1.1 O-symbols 1.2 Asymptotic sequences 1.3 Asymptotic expansions 1.4 Linear operations with asymptotic expansions 1.5 Other operations with asymptotic expansions 1.6 Asymptotic power series 1.7 Summation of asymptotic series References Chapter II. Integrals 2.1 Integration by parts 2.2 Laplace integrals 2.3 Critical points 2.4 Laplace's method 2.5 The method of steepest descents 2.6 Airy's integral 2.7 Further examples 2.8 Fourier integrals 2.9 The method of stationary phase References Chapter III. Singularities of Differential Equations 3.1 Classification of singularities 3.2 Normal solutions 3.3 The integral equation and its solution 3.4 Asymptotic expansions of the solutions 3.5 Complex variable. Stokes' phenomenon 3.6 Bessel functions of order zero References Chapter IV. Differential Equations with a Large Parameter 4.1 Liouville's problem 4.2 Formal solutions 4.3 Asymptotic solutions 4.4 Application to Bessel functions 4.5 Transition points 4.6 Airy functions 4.7 Asymptotic solutions valid in the transition region 4.8 Uniform asymptotic representations of Bessel functions References
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