1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

200 CHAPTER 6 • COMPLEX lNTECRATION

If C is a smooth curve, then ds, the differential of arc length, is given by

ds = V[x' (t)]^2 + [y' (t)]^2 dt = lz' (t)I dt.

The function s (t) = Vix' (t)]^2 + [y' (t)]^2 is continuous, as x' and y' are

continuous functions, so the length L ( C) of the curve C is

(6-11)

Now, consider C to be a curve with parametrization

C: z1 (t) =x(t)+iy(t), for a~ t ~ b.

The opposite curve -C traces out the same set of points in the plane, but

in the reverse order, and has the parametrization

-C: z2(t) = x(- t) + iy(- t) , for - b ~ t ~ - a.

Since z 2 (t) = z 1 (- t), - C is merely C traversed in the opposite sense, as

illustrated in Figure 6.3.

A curve C that is constructed by joining finitely many smooth curves end to

end is called a contou r. Let C 1 , C 2 , .•. , C,,. denote n smooth curves such that

the terminal point of the curve Ck coincides with the initial point of Ck+1 1 for

k = 1, 2, ... , n - 1. We express the contour C by the equation

A synonym for contour is p ath.

• EXAMPLE 6.5 Find a parametrization of t he polygonal path C from - 1 +i
  to 3 - i shown in Figure 6.4.

S olution We express C as three smooth curves, or C = C 1 + C2 + C3. If we

set z 0 = -1 + i and z 1 = -1, we can use Equation (1-48) to get a formula for

the straight-line segment joining two points:

C1: zi(t) = Zo + t(z1 -zo) = (- 1 + i) + t [-1-(-1 + i)], for 0 ~ t ~ 1.

When simplified, this formula becomes

C 1 : z1 (t) = - 1 + i (1 - t), for 0 ~ t ~ 1.

Figure 6 .3 The curve C and its opposite curve -C.