Functions and Analysis

2504 Submissions

[3] ai.viXra.org:2504.0104 [pdf] submitted on 2025-04-26 17:31:39

Sampling on the Riemann Surface: A Natural Resolution of Branch Cuts in Puiseux Series

Authors: Chang Hee Kim
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references)

This paper presents a novel yet profoundly natural approach to handling branch cuts in multivalued complex functions, especially Puiseux series, by sampling directly on the Riemann surface. Rather than imposing artificial branch cuts, we explore how FFT-based sampling across monodromy inherently respects the geometry of multivalued functions. The method aligns with the deep structure of Riemann surfaces and offers a unified computational framework for extracting coefficients in power series, including fractional exponents.
Category: Functions and Analysis

[2] ai.viXra.org:2504.0062 [pdf] replaced on 2025-08-01 00:04:03

Fine-Tuning the Generating Function Technique for Nonlinear Partial Differential Equations

Authors: Robert L. Jackson
Comments: 9 Pages. Obtain zip files via https://notebookarchive.org/2025-07-9qxwu8n. Contact information is rljacksonmd@gmail.com

This article emphasizes the fine-tuning step of the Generating Function Technique (GFT), a crucial component that enhances solution accuracy and computational efficiency for nonlinear partial differential equations (NPDEs). Unlike other methods, such as the Simplest Equation Method, G’/G-Expansion Method, Adomain Decomposition, and Homotopy Perturbative Method, the fine-tuning step within GFT systematically optimizes the solution series. This paper demonstrates the impact of fine-tuning through detailed applications to inhomogeneous NPDEs, elucidating its capability in generating superior analytical solutions.
Category: Functions and Analysis

[1] ai.viXra.org:2504.0027 [pdf] submitted on 2025-04-10 19:44:18

A Unified Computational Framework Unifying Taylor-Laurent, Puiseux, Fourier Series, and the FFT Algorithm

Authors: Chang Hee Kim
Comments: 30 Pages. English Includes C++ code in the appendix for demonstration Copyright License: Creative Commons Attribution-NonCommercial 4.0 International License

This paper proposes a unified framework for understanding and computing Taylor-Laurent Series, Puiseux Series, Fourier Series, and the Fast Fourier Transform (FFT) algorithm. By examining the connections between these seemingly disparate mathematical concepts, we show that they are not distinct, but manifestations of a single computational principle. We explore their shared foundation in series expansion, their computational embodiment via linear systems, and the central role of FFT as the universal engine.
Category: Functions and Analysis