1550251515-Classical_Complex_Analysis__Gonzalez_

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 1 the oriented angle a from the positive direction of

line L to the vector z - a is negative and greater than -7f i.e., -7f <

Arg(z - a)/b < 0. Hence (2.4-16) holds.

Similarly, for z E P 2 the oriented angle /3 from the positive direction of

L to the vector z -a is positive and less than 7f, i.e., 0 < Arg( z -a)/ b < 7f.

Hence (2.4-17) holds. Of course, for points z on the line L, we have z =

a+ bt, o:r (z - a)/b = t (a real number). Hence

Im z-a =0

b

for z EL (2.4-18)

If z = a' + b' r is another representation of the same oriented line L,
we must have

a' - a= μb and b' = >.b

where μ and >. are real and ).. > 0. Hence for z fi L we have

z ~'a' = z - ~; μb = ~ ( z ~ a _ μ)

so that

z -a' 1 z - a

Im--= -lm--

b' ).. b

which shows that lm(z-a')/b' and Im(z-a)/b have the same sign. There-

fore, the foregoing characterizations of P 1 and P 2 are independent of the
parametric representation of L as long as the orientation determined on
L remains the same.