Fabry gap theorem
In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence.
The theorem may be deduced from the first main theorem of Turán's method.
Statement of the theorem
Let 0 < p1 < p2 < ... be a sequence of integers such that the sequence pn/n diverges to ∞. Let j∈N be a sequence of complex numbers such that the power serieshas radius of convergence 1. Then the unit circle is a natural boundary for the series f.