Derivative and higher-order Cauchy integral formula of matrix functions

The derivative of a n-order matrix function on the complex field is usually defined as a n(2)-order matrix, which is not suitable for generalizing Cauchy integral formula of matrix functions to its higher-order derivative form. In this paper, a new kind of derivative of matrix functions is defined, and the higher-order derivative form of Cauchy integral formula of matrix functions is also proved to be true under the new kind of definition of derivative. At the same time, some examples about calculating the values of matrix functions by using Cauchy integral formula of matrix functions and its higher-order derivative formare given.

Automated summary

A new type of derivative of matrix functions is defined in this paper, and it is proven that the higher-order derivative form of the Cauchy integral formula for matrix functions holds under this new definition. Additionally, examples of calculating matrix function values using the Cauchy integral formula and its higher-order derivative form are provided.