Banishing divergence Part 2: Limits of oscillatory sequences, and applications (1108.4952v1)

Abstract: Sequences diverge either because they head off to infinity or because they oscillate. Part 1 \cite{Part1} of this paper laid the pure mathematics groundwork by defining Archimedean classes of infinite numbers as limits of smooth sequences. Part 2 follows that with applied mathematics, showing that general sequences can usually be converted into smooth sequences, and thus have a well-defined limit. Each general sequence is split into the sum of smooth, periodic (including Lebesgue integrable), chaotic and random components. The mean of each of these components divided by a smooth sequence, or the mean of the mean, will usually be a smooth sequence, and so the oscillatory sequence will have at least a leading term limit. Examples of limits of oscillatory sequences with well-defined limits are given. Methodologies are included for a way to calculate limits on the reals and on complex numbers, a way to evaluate improper integrals by limit of a Riemann sum, and a way to square the Dirac delta function.

• The paper presents a method to define limits for traditionally divergent or oscillatory sequences by decomposing and transforming them into smooth sequences.
• The core methodology involves representing general sequences through component decomposition and statistical evaluation of transformed components to yield well-defined limits.
• Applications include finding limits of complex functions, evaluating improper integrals, and manipulating generalized functions like the Dirac delta function.

Analyzing Divergence Elimination in Oscillatory Sequences and Applications

The paper "Banishing divergence Part 2: Limits of oscillatory sequences, and applications" by David Alan Paterson addresses the challenge of defining limits for sequences that traditionally exhibit divergence due to either unbounded growth or oscillatory behavior. It builds on the groundwork of its predecessor by expanding the concept of finite limit determination from smooth sequences to a broader class of general sequences, including those with significant oscillatory characteristics. The central thesis is the decomposition of any general sequence into the sum of four types of components—smooth, periodic, chaotic, and random—and demonstrating how transformed versions of these sequences often yield smooth sequences with well-defined limits.

Fundamental Concepts and Methodology

The work begins by laying down significant mathematical conceptions introduced in Part 1, namely, the representation of sequences in terms of infinite ordinal numbers and the establishment of smooth sequences. Paterson escalates this by illustrating that general sequences can be systematically transformed such that they relate to smooth sequences, ultimately leading to limits that are determinable. This conversion of oscillatory components to smooth sequences involves statistically evaluating components through their mean values, a technique analogous yet distinct from traditional approaches like Cesàro summation.

Application and Implications

The paper provides robust examples, demonstrating the efficacy of this methodology in various contexts, including the limits of complex functions, evaluation of improper integrals, and the manipulation of generalized functions such as the Dirac delta function. In particular, applications described are of theoretical significance:

• Finding the limit of functions at points of discontinuity in the real number domain.
• Employing nonstandard contour integration for complex functions.
• Utilizing adapted Riemann sums to assess improper integrals.
• Squaring the Dirac delta function, which is usually a mathematically non-trivial operation due to its singular nature.

These methodologies emphasize the transformation of non-smooth sequences into a smooth form before limit application, preserving the locality of limit definitions and facilitating the evaluation independently of other values in the sequence.

Theoretical and Practical Implications

The theoretical implications of this work are profound, encouraging mathematicians to rethink the concepts of divergence and convergence beyond classical paradigms. By potentially establishing that every sequence has a well-defined leading term limit and every definite integral can be expressed within the devised smooth framework, Paterson introduces new scope for mathematical analysis that could impact a wide range of mathematical and engineering applications.

Practically, the techniques outlined could substantially benefit computational approaches, offering streamlined processes for calculating limits that previously required labor-intensive or approximate methods. Speculatively, development into software could automate these processes, allowing for instantaneous evaluation of complex limits and integral solutions.

Conclusion

Ultimately, this paper extends the mathematical toolkit available for handling divergent sequences and demonstrates practical methodologies to extract meaningful limits from sequences previously considered resistant to such analysis. The work opens new avenues for both theoretical exploration and practical computational advancements, encouraging further investigation into its broader implications and potential generalizations. Researchers are invited to explore the conjectures proposed, potentially paving the way for further breakthroughs in understanding oscillatory and divergent sequences.