Lacunary function

In analysis, a lacunary function or 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 lacuna (pl. lacunae), meaning gap, or vacancy.

The first examples of lacunary functions involved Taylor series with large gaps, or lacunae, between the non-zero coefficients of their expansions. More recent investigations examine Fourier series with similar gaps between non-zero coefficients. In modern usage, a lacunary series may be either a Taylor series or a Fourier series.