 The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by

Also, if a complex analytic function is defined in an open ball around a point "x"0, its power series expansion at "x"0 is convergent in the whole open ball (holomorphic functions are analytic).This statement for real analytic functions (with open ball meaning an open interval of the real line rather than an open disk of the complex plane) is not true in general; the function of the example above gives an example for "x"0 = 0 and a ball of radius exceeding 1, since the power series diverges for |"x"| > 1.Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane.