Multicomplex Taylor Series Expansion for Computing High-Order Derivatives

Multicomplex Taylor series expansion (MCTSE) is a numerical method for calculating higher-order partial derivatives of a multivariable real-valued and complex-valued analytic function based on Taylor series expansion without subtraction cancelation errors. The implementation has been facilitated using Cauchy-Riemann matrix representation of multicomplex variables. In this paper, we show steps for finding these matrices and, in addition, that the number of appearances of the k-th derivatives follows the Pascal's triangle. Also, the situations where the MCTSE is not applicable is determined. Finally, we investigate the application of the method for complex-valued functions.