1550251515-Classical_Complex_Analysis__Gonzalez_

88 Chapter^2

In either case the graph of the circle is the set of points (2.4-20).
( c) An· oriented circular arc with center at a and radius r is defined by
(2.4-24)

In Section 3.13 we consider in some detail the concept of arc in general.

11. Inside and outside a circle. Clearly, the inequality

lz - al< r (2.4-25)

determines the set of points z lying inside the circle with center at a and

radius r. Similarly, the inequality

lz - al> r (2.4-26)

determines the set of points z that are outside the same circle.

12. Angular region. The angular region bounded by the rays Arg z =a
    and Arg z = /3 (a < /3) is the set of points A defined as follows:

A= {z: a:::; Argz:::; /3} U {O} (2.4-27)

13. The principal region. The set of points

Cp = {z: lzl +z =f. O} (2.4-28)

is referred to as the principal region. It coincides with the complement of

nonpositive real numbers.

14. The punctured plane. This is the set

C(zo) = C - {zo} = {z: lz - zol =f. O} (2.4-29)

Often the deleted point is z 0 = 0.

15. The cut plane. This is a triplet {A,g,h} composed of the open
    angular region

A = { z: -?T < arg z < +1T}

and two mappings g: z = t, -oo < t :::; 0, and h: z = -t, 0 :::; t < +oo,
defining an initial ray and a terminal ray, their orientations being deter-
mined by the respective parametrizations. Intuitively, the plane is supposed
to be cut along the negative real axis and the two resulting boundaries are
distinguished by their orientations (Fig. 2.6). The plane may also be cut
from a point zo to oo along some specified direction.

• -^0 x
  Fig. 2.6