Embarking on Adventurous Journeys Through Some Subjects in Pure Mathematics: Part II

As we enter Part II, we turn to two new subjects: Extensions of Fundamental Trigonometric and Hyperbolic Functions and Unraveling the Mystery of the Riemann Hypothesis: Toward a Complete Proof, beginning with Chapter 3 and Chapter 4.Chapter 3 develops generalized forms of classical trigonometric and hyperbolic functions, elevating them to a broader conceptual framework. These extended formulations introduce new dynamic behaviors, enabling the reshaping of familiar functions and the exploration of entirely new families of elementary functions. This extension creates a richer mathematical framework for studying natural curved surfaces. Classical trigonometric functions such as sin(x) and cos(x) produce identical cross sections when plotted in three dimensions, yielding surfaces that replicate the same curve for every value of y. In contrast, the new two variable trigonometric functions generate curves that vary with y, producing surfaces that differ across the domain and giving rise to distinct geometric objects rather than simple extrusions.Chapter 4 presents a proposed proof of the Riemann Hypothesis, one of the most intriguing and significant problems in mathematics. We explore known findings and properties of the Riemann zeta function, reformulate the functional equation in a new light, and demonstrate why certain hypothetical pairs of nontrivial zeros cannot exist. We present a proof of the Riemann Hypothesis using an elementary algebraic approach. In Section 4-4, we demonstrate that all possible nontrivial zeros lie in the narrow bands 6.28318534... ≤ 6.28983598... or -6.28983598... ≤ Im(s)