Lacunary function
In analysis, a lacunary function, also known as a lacunary series, is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined by a power series. The word lacunary is derived from (pl. lacunae), meaning gap, or vacancy.
The first known examples of lacunary functions involved Taylor series with large gaps, or lacunae, between the non-zero coefficients of their expansions. More recent investigations have also focused attention on Fourier series with similar gaps between non-zero coefficients. There is a slight ambiguity in the modern usage of the term lacunary series, which may be used to refer to either Taylor series or Fourier series.
Consider the lacunary function defined by a simple power series:
The power series converges uniformly on any open domain |z| < 1. This can be proved by comparing f with the geometric series, which is absolutely convergent when |z| < 1. So f is analytic on the open unit disk. Nevertheless, f has a singularity at every point on the unit circle, and cannot be analytically continued outside of the open unit disk, as the following argument demonstrates.
Clearly f has a singularity at z = 1, because
is a divergent series. But since
we can see that f has a singularity at a point z when z2 = 1 (that is, when z = ±1), and also when z4 = 1 (that is, when z = ±1 or when z = ±i). By the induction suggested by the above equations, f must have a singularity at each of the 2nth roots of unity for all natural numbers n. The set of all such points is dense on the unit circle, hence by continuous extension every point on the unit circle must be a singularity of f.
Evidently the argument advanced in the simple example can also be applied to show that series like
also define lacunary functions. What is not so evident is that the gaps between the powers of z can expand much more slowly, and the resulting series will still define a lacunary function. To make this notion more precise some additional notation is needed.
...
Wikipedia