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