WWGD said:Are you interested in the more theoretical or more practical side, i.e., how to use non-standard analysis or the theory part? Or both?
Non-standard analysis is a mathematical theory that extends the traditional framework of calculus and analysis to include infinitesimals and infinitely large numbers. It was first developed by Abraham Robinson in the 1960s as a way to rigorously define and manipulate these non-standard quantities.
Non-standard analysis is based on a different foundation than traditional analysis. While traditional analysis uses the epsilon-delta definition of limits, non-standard analysis uses a framework called the hyperreal numbers, which includes infinitesimal and infinite numbers. This allows for a more intuitive and powerful approach to calculus and analysis.
Non-standard analysis has been applied in various fields such as physics, biology, economics, and computer science. It has also been used to solve problems in traditional analysis, such as the construction of non-measurable sets and the solution to the Basel problem.
While non-standard analysis provides a powerful tool for solving problems in analysis, it is not without its limitations. Some argue that it is not as widely accepted as traditional analysis and may not be applicable to all mathematical problems. Additionally, the use of infinitesimals and infinite numbers may be seen as non-intuitive by some mathematicians.
There are many books available on non-standard analysis, both for beginners and for more advanced readers. Some popular titles include "Nonstandard Analysis" by Martin Davis, "Elementary Calculus: An Infinitesimal Approach" by H. Jerome Keisler, and "Nonstandard Methods in Stochastic Analysis and Mathematical Physics" by Sergio Albeverio and Raphael Hoegh-Krohn. These books can be found in most major bookstores or online retailers.