Logarithmic and Hyperbolic Structures in Mathematics and ‎Their Conceptual Analogies in Physics

James Russell Farmer ‎

doi:10.55672/hij2026pp19-24

Authors

James Russell Farmer ‎ Independent Researcher, 10 William Ave, Greenlane, Auckland 1051, New Zealand

DOI:

https://doi.org/10.55672/hij2026pp19-24

Keywords:

logarithmic integral, asymptotic behavior, mathematical physics, relativistic structures, singularities

Abstract

The inverse function occupies a special position in elementary calculus because the standard power-law rule for integration ‎fails for a particular exponent. This exceptional behavior motivates a broader investigation into the mathematical structure of ‎logarithmic and hyperbolic functions. In this article we examine the asymptotic behavior of logarithmic and inverse functions, ‎with particular emphasis on singularities, asymptotes, and intersection structure. We investigate how logarithmic behavior ‎naturally emerges from inverse power relationships and discuss the geometric significance of the transition between these two ‎classes of functions. The discussion is then extended toward conceptual analogies in physics, especially in relation to ‎asymptotic behavior appearing in special relativity and gravitational theory. Rather than proposing modifications to ‎established physical theories, the article explores mathematical parallels between divergent structures, limiting processes, and ‎physical interpretation.‎ The aim of this work is therefore not to replace existing physical theories, but to highlight mathematical patterns that recur in ‎both analysis and theoretical physics.‎

Downloads

Download data is not yet available.

References

‎[1] Stewart, J. (2015). Calculus: Early Transcendentals (8th ‎ed.). Cengage Learning.‎

‎[2] Apostol, T. M. (1967). Calculus, Volume 1. Wiley.‎

‎[3] Courant, R., & John, F. (1999). Introduction to Calculus ‎and Analysis, Volume 1. Springer.‎

‎[4] Arfken, G. B., Weber, H. J., & Harris, F. E. (2013). ‎Mathematical Methods for Physicists (7th ed.). Academic ‎Press.‎

‎[5] Bender, C. M., & Orszag, S. A. (1999). Advanced ‎Mathematical Methods for Scientists and Engineers. ‎Springer.‎

‎[6] Einstein, A. (1905). On the electrodynamics of moving ‎bodies. Annalen der Physik, 17, 891–921.‎

‎[7] Einstein, A. (1916). The foundation of the general theory ‎of relativity. Annalen der Physik, 49, 769–822.‎

‎[8] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., ‎& Knuth, D. E. (1996). On the Lambert W function. ‎Advances in Computational Mathematics, 5, 329–359.‎

‎[9] Farmer, J. R., & Musakhail, M. A. (2026). An exploratory ‎framework for gravitation and electrodynamics: A ‎Lagrangian-Hamiltonian perspective. Hyperscience ‎International Journal (HIJ), 1–12. ‎https://doi.org/10.55672/hij2026pp1-12‎

‎[10] Farmer, J. R., & Musakhail, M. A. (2026). Lagrangian ‎dynamics of the Musakhail Aether Dynamical ‎Lagrangian. Hyperscience International Journal (HIJ), ‎‎13–18. https://doi.org/10.55672/hij2026pp13-18‎

‎[11] Schutz, B. (2009). A First Course in General Relativity ‎‎(2nd ed.). Cambridge University Press (2008)‎