Stolz Angle

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The Demonstration shows a symmetrical circular sector inside the unit disk in the complex plane, with the vertex at the point 1. The angle enclosed by such a sector is known as a Stolz angle. It also shows, for various constants , the region in the unit disk where the relationship holds. We see that the interior of every Stolz angle is contained in some and every is contained in some Stolz angle (e.g. the one determined by the two tangents to the boundary of at 1). You can create arbitrary Stolz angles with the vertex at 1 by dragging its upper corner point inside the unit disk. The Stolz angle and the region play a key role in the complex version of Abel’s limit theorem (described in Details).

Contributed by: Andrzej Kozlowski (March 2011)
Open content licensed under CC BY-NC-SA



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.

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