Investigating Impact of Parameters on Hyperbolic Function Generalization

Generalization of the ordinary hyperbolic functions called hyperbolic Ateb-functions is considered. They are the inverse of incomplete Beta-function. These functions are solutions of differential equations, which describe the aperiodic vibration motion. It is shown that hyperbolic Ateb-functions have different dependence levels on their parameters. Investigation of the domain of hyperbolic Ateb-functions is conducted. It is shown that the minimum value of the domain can be expressed in terms of the lemniscate constant. The formulas for derivatives of hyperbolic Ateb-functions are proved and the structure of the higher-order derivatives is obtained. Some other properties connected with symmetry are considered. Taylor expansions of Ateb-cosine and Ateb-sine are taken out. Based on the mathematical induction principle, the corresponding theorems are proven. Examples of Taylor expansions of Ateb-cosine and Ateb-sine for different parameters are presented. The comparison of Ateb-function calculation using Taylor expansion and numerical methods shows the advantage of the Taylor series approach.