1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 85 To see clearly why we distinguish between the mapping (which we call the oriented closed seg ...

86 x Fig. 2.4 and the left half-plane P 2 as the set of points z such that z-a Im-b->0 Chapter 2- (2.4-17) In fact, for z E P ...

Topology of Plane Sets of Points 87 Circles. (a) Since lz - al expresses the distance from a to z, ·the equation of a circle C( ...

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 ...

Topology of Plane Sets of Points 89 EXERCISES 2.2 Describe graphically the sets of points in the complex plane defined by the fo ...

90 or alternatively, z = S(z) = z1 - z2 z1z2 - z1z2 ---z+----- z1 - z2 z1 - z2 Chapter 2 The function S(z) on the right-hand sid ...

Topology of Plane Sets of Points 91 (S, d) consisting of a nonempty set S and a distance d defined on S x S, · i.e., defined for ...

92 Chapter 2 The extended complex number system C with the chordal metric as defined in (1.17-14), (1.17-15), and with x( oo, oo ...

Topology of Plane Sets of Points 93 is also a metric space, denoted 02 [a, b]. It is called the space of continuous functions wi ...

94 Chapter^2 v y (^0) x 0 x (a) (b) Fig. 2.7 Definition 2.8 For any 8 > 0 the set N 0 (a) = {x: x E S,d(x,a) < 8} is calle ...

Topology of Plane Sets of Points 95 If N is a neighborhood of a, the set N' = N - {a} is called a deleted neighborhood of a. In ...

The intersection of a finite number of open subsets is open. The union of a finite number of closed sets of a metric space is c ...

Topology of Plane Sets of Points Example Consider in ( C, d) 'the set A= {z: JzJ < 1} U {z: z^2 = -1} Then Int A= {z: Jzl < ...

98 Chapter^2 Definition 2.21 The set of all limit points of a set A C S is called the derived set of A, denoted A or A°'. In the ...

Topology of Plane Sets of Points 99 Thus the diameter of A is the supremum (least upper bound) of the set of all distances betwe ...

100 Chapter^2 9. If A is a bounded set in (S, d), show that A is also a bounded set. 10. Show that A C Sis nowhere dense in S if ...

Topology of Plane Sets of Points 101 To see that the condition is not necessary, consider in C the sets A = {z: lzl < 1} and ...

102 Chapter 2 which is a representation of A as the union of two disjoint nonempty closed sets with respect to the subspace A. S ...

Topology of Plane Sets of Points 103 Intervals of the form [a, a] are clearly connected, and the argument used above for JR can ...

104 Chapter^2 the polygonal line, call it I, must join a point in G 1 to a point in G2. Now and since G 1 and G 2 are separated, ...