 However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane.The function ƒ("x") defined in the paragraph above is a counterexample, as it is not defined for "x" = ±"i".This explains why the Taylor series of ƒ("x") diverges for |"x"| > 1, i.e., the radius of convergence is 1 because the complexified function has a pole at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.