Analytic Function - Real Versus Complex Analytic Functions

Real Versus Complex Analytic Functions

Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.

According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. 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 x₀, its power series expansion at x₀ is convergent in the whole ball (analyticity of holomorphic functions). 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 x₀ = 0 and a ball of radius exceeding 1, since the power series 1 − x2 + x4 − x6... 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. 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.