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Formalizing Mechanical Analysis Using Sweeping Net Methods II: Written Without Complex Analysis and With Complex Analysis

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Title: Formalizing Mechanical Analysis Using Sweeping Net Methods II: Written Without Complex Analysis and With Complex Analysis
Authors: Emmerson, Parker
Publisher Information: Zenodo, 2024.
Publication Year: 2024
Subject Terms: Meromorphic Functions, Analytic Functions, Complex Analysis, Boundary Behavior, Conformal Mappings, Sweeping Net Methods, Angular Limits, Open Unit Disk, Fourier Series, Topological Equivalence, Harmonic Analysis, Argument Analysis, Mapped Singularities, Computational Visualization, Riemann Mapping Theorem, Analytic Continuation, Singular Integral Equations, Modulus on Circle, Schwarz Reflection Principle, Cauchy Integrals, Maximum Modulus Principle, Unit Circle, Parametrization, Theorems, Continuous Functions, Cluster Sets, Singularity Approximation, Matplotlib, Function Behavior, Blaschke Product, Winding Numbers, Numpy, Essential Singularities, Poles, Singularities, Discretization, Python, Argument of Function
Description: Published with great thanksgiving for Yaohushua, the living One Yahowah, "Jesus Christ." In previous work, Formalizing Mechanical Analysis Using Sweeping Net Methods I, sweeping netmethods have been extended to complex analysis, relying on the argument of complex functions definedon the unit circle. In this paper, we reformulate these methods purely within a real-valued and geometricframework, avoiding the use of complex analysis. By redefining the sweeping net constructs and theassociated theorems using real functions and geometric interpretations on the unit circle, we demonstratehow singularities and their approximations can be effectively analyzed without the need for imaginarynumbers. This approach provides intuitive geometric insights and broadens the applicability of sweepingnet methods in mathematical analysis.
Document Type: Article
Language: English
DOI: 10.5281/zenodo.13968738
DOI: 10.5281/zenodo.13968739
Rights: CC BY
Accession Number: edsair.doi.dedup.....44f9e3de9ab1c9666d5a2e6f0448b5e8
Database: OpenAIRE
Description
Abstract:Published with great thanksgiving for Yaohushua, the living One Yahowah, "Jesus Christ." In previous work, Formalizing Mechanical Analysis Using Sweeping Net Methods I, sweeping netmethods have been extended to complex analysis, relying on the argument of complex functions definedon the unit circle. In this paper, we reformulate these methods purely within a real-valued and geometricframework, avoiding the use of complex analysis. By redefining the sweeping net constructs and theassociated theorems using real functions and geometric interpretations on the unit circle, we demonstratehow singularities and their approximations can be effectively analyzed without the need for imaginarynumbers. This approach provides intuitive geometric insights and broadens the applicability of sweepingnet methods in mathematical analysis.
DOI:10.5281/zenodo.13968738