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

converges then

tends to

as long as

approaches 1 in such a way that

stays in the region

in which

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 "

moves within

" imply each other.

[1] L. V. Ahlfors,

*Complex Analysis,* 3rd ed., New York: McGraw–Hill, 1979.