1550251515-Classical_Complex_Analysis__Gonzalez_

82 Chapter 2

so it has slope m 1 = -a//3, if /3 :f: 0. Since the slope of A is m 2 = /3 /a

(assuming that a :f: 0), we see that m 1 m2 = -1. If either a = 0 or

/3 = 0, one of the lines is horizontal while the other is vertical. The case

a = /3 = 0 cannot occur because A :f: 0.

2. Oriented straight line. The mapping defined by the single-valued

function z =a+ bt, -oo < t < +oo, where a and bare complex numbers,

b :f: 0, and t is a real parameter, will be called an oriented straight line. It

is so called because the range of the function, i.e., the set of points

{z: z=a+bt,b:f=O,tEJR} (2.4-2)

lies in the straight line L of the complex plane that passes through the

point a and has the direction of the vector b (Fig. 2.2), as follows from

the geometric interpretation of addition and the fact that the vector bt, for

t :f: 0 real, has the same direction (possibly not the same orientation) as

vector b. Note that Arg(bt) = Arg b + Arg t + 2mr, where Arg t = 0 or 7r

and n = -1, 0, or 1.

It is said that the mapping defines an oriented straight line because the

usual ordering of the real numbers induces a certain ordering of the points

of L, called the natural order, the positive orientation or the orientation

induced by the parametrization, which is defined as follows. If z' = a + bt',

z^11 = a + bt", and t' < t", we consider z' as preceding z^11 • This may be

indicated by writing z' -< z". The opposite convention about the ordering

of the points of the line L will be called the negative orientation of the line.

Since ;^11 - z' = b( t" -t'), if t' < t", we have t" -t' > 0. Hence the vector

z" - z' has the same orientation as vector b. Thus the positive orientation

of an oriented line L defined by z = a+ bt coincides with the orientation of

vector b. This is sometimes indicated by an arrowhead on the line or by a

small arrow drawn parallel to L in the appropriate sense. Clearly, t' ·:f= t"

y L

z = a+ bt

x

Fig. 2.2