Original sentence: Cauchy's integral formula states that for any analytic function  defined on a set  (), it holds.If the matrix  is diagonalizable,.If the real series converges for , then the corresponding matrix series will converge for matrix argument  if  for some matrix norm  which satisfies.Now replace  by a matrix  and consider a path  inside  that encloses all eigenvalues of.One possibility to achieve this to let  be a circle around the origin with radius  for an arbitrary matrix norm.then a matrix function can be defined by substituting  by a matrix: the powers become matrix powers, the additions become matrix sums and the multiplications become scaling operations.where  is a closed curve inside the domain  enclosing
Edited sentence: Cauchy's integral formula states that for any analytic function f defined on a set D (), it holds.If the matrix A is diagonalizable,.If the real series converges for , then the corresponding matrix series will converge for matrix argument A if  for some matrix norm  which satisfies.Now replace x by a matrix A and consider a path C inside D that encloses all eigenvalues of A.One possibility to achieve this to let C be a circle around the origin with radius  for an arbitrary matrix norm.then a matrix function can be defined by substituting x by a matrix: the powers become matrix powers, the additions become matrix sums and the multiplications become scaling operations.where C is a closed curve inside the domain D enclosing x