Stolz Angle

show the region

1-z

1-z

≤K

K

1.5

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

K>1

, the region

R(K)

in the unit disk where the relationship

1-z

1-z

≤K

holds. We see that the interior of every Stolz angle is contained in some

R(K)

and every

R(K)

is contained in some Stolz angle (e.g. the one determined by the two tangents to the boundary of

R(K)

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

R(K)

play a key role in the complex version of Abel’s limit theorem (described in Details).