Nonstandard Analysis in Practice [Paperback]

This book introduces the graduate mathematician and researcher to the effective use of nonstandard analysis (NSA). It provides a tutorial introduction to this modern theory of infinitesimals, followed by nine examples of applications, including complex analysis, stochastic differential equations, differential geometry, topology, probability, integration, and asymptotics. It ends with remarks on teaching with infinitesimals.

1. Tutorial.- 1.1 A new view of old sets.- 1.1.1 Standard and infinitesimal real numbers, and the Leibniz rules.- 1.1.2 To be or not to be standard.- 1.1.3 Internal statements (standard or not) and external statements.- 1.1.4 External sets.- 1.2 Using the extended language.- 1.2.1 The axioms.- 1.2.2 Application to standard objects.- 1.3 Shadows and S-properties.- 1.3.1 Shadow of a set.- 1.3.2 S-continuity at a point.- 1.3.3 Shadow of a function.- 1.3.4 S-differentiability.- 1.3.5 Notion of S-theorem.- 1.4 Permanence principles.- 1.4.1 The Cauchy principle.- 1.4.2 Fehrele principle.- 2. Complex analysis.- 2.1 Introduction.- 2.2 Tutorial.- 2.2.1 Proof of the Robinson-Callot theorem.- 2.2.2 Applications.- 2.2.3 Exercises with answers.- 2.2.4 Periodic functions.- 2.3 Complex iteration.- 2.4 Airys equation.- 2.4.1 The distinguished solutions.- 2.5 Answers to exercises.- 3. The Vibrating String.- 3.1 Introduction.- 3.2 Fourier analysis of (DEN).- 3.2.1 Diagonalisation of A.- 3.2.2 Interpretation of N i-large.- 3.2.3 Resolution of (DEN).- 3.3 An interesting example.- 3.4 Solutions of limited energy.- 3.4.1 A preliminary theorem.- 3.4.2 Limited energy: S-continuity of solution.- 3.4.3 Limited energy: propagation and reflexion.- 3.4.4 A particular case: comparison with classical model.- 3.5 Conclusion.- 4. Random walks and stochastic differential equations.- 4.1 Introduction.- 4.2 The Wiener walk with infinitesimal steps.- 4.2.1 The law of wt for a fixed t.- 4.2.2 Law of w.- 4.3 Equivalent processes.- 4.3.1 The notion.- 4.3.2 MacroscoplC9