Hyperbolic complex numbers in two dimensions

In this study, general information about two-dimensional hyperbolic complex numbers is provided. The logarithmic and trigonometric expressions for a hyperbolic complex number are discussed. Additionally, the properties of the two complex-valued functions are presented. The two-complex valued functions defined by power series and their analyticity are given. Furthermore, it is shown that there are equality relations between the partial derivatives of the hyperbolic two-complex valued real component functions. It is also noted that the integral of the two complex functions between two points is independent of the path connecting the points. On the other hand, despite not being able to be separated into factors, a hyperbolic two-complex polynomial can be written as a linear or quadratic product of factors.