Author | Estrada, Ricardo. author |
---|---|
Title | Asymptotic Analysis [electronic resource] : A Distributional Approach / by Ricardo Estrada, Ram P. Kanwal |
Imprint | Boston, MA : Birkhรคuser Boston, 1994 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0029-8 |
Descript | IX, 258 p. online resource |
1 Basic Results in Asymptotics -- 1.1 Introduction -- 1.2 Order Symbols -- 1.3 Asymptotic Series -- 1.4 Algebraic and Analytic Operations -- 1.5 Existence of Functions with a Given Asymptotic Expansion -- 1.6 Asymptotic Power Series in a Complex Variable -- 1.7 Asymptotic Approximation of Partial Sums -- 1.8 The Euler-Maclaurin Summation Formula -- 2 Introduction to the Theory of Distributions -- 2.1 Introduction -- 2.2 The Space of Distributions $$\mathcal{D}'$$ -- 2.3 Algebraic and Analytic Operations -- 2.4 Regularization, Pseudofunction and Hadamard Finite Part -- 2.5 Support and Order -- 2.6 Homogeneous Distributions -- 2.7 Distributional Derivatives of Discontinuous Functions -- 2.8 Tempered Distributions and the Fourier Transform -- 2.9 Distributions of Rapid Decay -- 2.10 Spaces of Distributions Associated with an Asymptotic Sequence -- 3 A Distributional Theory of Asymptotic Expansions -- 3.1 Introduction -- 3.2 The Taylor Expansion of Distributions -- 3.3 The Moment Asymptotic Expansion -- 3.4 Expansions in the Space $$\mathcal{P}'$$ -- 3.5 Laplaceโs Asymptotic Formula -- 3.6 The Method of Steepest Descent -- 3.7 Expansion of Oscillatory Kernels -- 3.8 The Expansion of f(?x) as ? โ> ? in Other Cases -- 3.9 Asymptotic Separation of Variables -- 4 The Asymptotic Expansion of Multidimensional Generalized Functions -- 4.1 Introduction -- 4.2 Taylor Expansion in Several Variables -- 4.3 The Multidimensional Moment Asymptotic Expansion -- 4.4 Laplaceโs Formula -- 4.5 Fourier Type Integrals -- 4.6 Further Examples -- 4.7 Tensor Products and Partial Asymptotic Expansions -- 4.8 An Application in Quantum Mechanics -- 5 The Asymptotic Expansion of Certain Series Considered by Ramanujan -- 5.1 Introduction -- 5.2 Basic Formulas -- 5.3 Lambert Type Series -- 5.4 Distributionally Small Sequences -- 5.5 Multiple Series -- 6 Series of Dirac Delta Functions -- 6.1 Introduction -- 6.2 Basic Notions -- 6.3 Several Problems That Lead to Series of Deltas -- 6.4 Dual Taylor Series as Asymptotics of Solutions of Differential Equations -- 6.5 Singular Perturbations -- References