Abel's power series theorem states that for every power series
with complex coefficients, there exists a number
, such that the series converges absolutely for
and diverges for
. The theorem does not say anything about the behavior of the series on the boundary of the disk.
There is a second theorem of Abel (called Abel's limit theorem by Lars Ahlfors) that refers to the case when we know that the series converges at a point on the boundary. Without loss of generality one can assume that
and the point is 1. In that case, one can show that if
as long as
approaches 1 in such a way that
stays in the region
is bounded by a fixed constant
Another way to state the complex Abel's limit theorem is to repeat the last sentence, but to replace "the region
" by "a Stolz angle"—a symmetrical circular segment with the vertex at 1 contained in the unit disk.
This Demonstration shows that the two formulations are equivalent. Namely, for every Stolz angle there exists a
such that the sector is contained in the corresponding region where the inequality is satisfied and, conversely, every
is contained in a Stolz angle. Thus the statements "
moves within a Stolz angle" and "
" imply each other.
 L. V. Ahlfors, Complex Analysis,
3rd ed., New York: McGraw–Hill, 1979.