Generalised p-Trigonometric and Hyperbolic Special Functions with Applications

Abstract

The p-trigonometric functions are a generalization of classical trigonometric functions, including sine, cosine, and tangent, as well as their inverse functions. These generalized functions have wide applications in various scientific fields, such as mathematical and physical modeling, and are relevant in numerous branches of science and engineering. This study investigates the identities and unique properties of p-trigonometric and inverse p-trigonometric functions, providing surface plots, locus plots, and proofs for derivative and integral formulas. Additionally, We explore p-hyperbolic functions, which extend classical hyperbolic functions, and examine their identities and properties, including inverse functions and their derivatives and integrals. Simulations are used to analyze the behavior of p-hyperbolic and inverse p-hyperbolic functions, demonstrating their utility in modeling complex data patterns. Moreover, the study extends p-trigonometric functions, such as sinp and cosp, into complex domains, and p-hyperbolic functions, including sinhp and coshp, into hyperbolic complex domains. We also investigate the connections between p-trigonometric and p-hyperbolic functions with imaginary arguments, revealing new properties and identities. The research further bridges hyperbolic and elliptical complex numbers to explore logarithmic functions with complex arguments. A special class of p-trigonometric functions is introduced, including p-versine, p-coversine, p-haversine, and p-hacovercosine, expanding on the established p-sine and p-cosine functions. This work offers new mathematical insights into these functions and their potential applications in diverse scientific domains, from mathematics to physics and engineering. Finally, We introduce a generalized Fourier series based on p-trigonometric functions, designed to represent periodic signals as sums of p-sine and p-cosine functions. By presenting integral formulas for products of these functions and demonstrating their orthogonality, We derive coefficients for the generalized p-Fourier series and provide examples. This series is applicable for expanding arbitrary forcing functions in solving non-homogeneous linear ordinary differential equations with constant coefficients, contributing valuable insights for future research in applied mathematics.